There are 230 space groups, belonging to 32 point groups. A space group is centrosymmetric if it contains the symmetry elements \bar{1} (= number "1" with a dash above it). It is sufficient to check whether the corresponding point group has this center of symmetry - then this is valid also for the space group. If you cannot do this because you are not familiar with space group/point group elations, I recommend the "International Tables", volume A (check e.g. http://www.iucr.org/__data/assets/pdf_file/0010/16597/ita.pdf). If the space group contains the symmetry element \bar{1}, it is centrosymmetric, and vice versa. For the 2 examples given under the link above, on page 108 you have symmetry operations for 6mm: 1, 3, 6, m (I omit +/- and different orientations). This means 6mm IS NOT centrosymmetric. For P2_1/c (page 184) you find operations: 1, 2, \bar{1}, c. So you have \bar{1}, and this space group IS centrosymmetric.

In the old version of Int.Tables at least one could check at the end, where the mathematics were given, whether the B coefficent is zero or not. This may also be of importance when there are different choices of origin and you have to adapt to a certain program for calculations. Beware!

We already had a lot of discussion about this in Reseachgate and you read them. It is not easy to differentiate. You can try the following: I(1) f you have a complete data set then do the intensity statistics. Read for example, M.M. Wolfson, Introduction to X-ray crystallography, p. 234., (2) You can use some physical properties of the crystal like ferroelectricity and second harmonic generation (SHG). If your crystal is ferroelectric then it must belong to a non-centrosymmetric space group. I can infer also the non-centrosymmetry by SHG.

If your question was whether a certain space group like P63cm (No. 185) then just go to the International Tables of Crystallography and check the symmetry elements. If -x,-y,-z is there then it is centrosymmetric and if not it is non-centrosymmetric. It is as simple as that. That is trivial. But if you question is whether your crystal belong to a centro- or non-centrosymmetric space group then read the answer I gave before. Mind that it is not easy to conclude. In case of doubt try centrosymmetric first and do the refinement and check whether that works. If that works then do not try non-centrosymmetric unless you have definite evidence from other physical properties.

According to Friedel's law regardless of whether the crystal structure is centro-symmetric or not, in the absence of the anomalous scattering the reciprocal space is always centro-symmetric

Because |F(hkl)| = |F(-h-k-l)| and Intensity of peak is square of |F|.

Thus in absence of anomalous scattering the absolute values of structure amplitudes and intensities is similar for both centro-symmetric and non-centro-symmetric space group. So by any diffraction experiment it will be difficult to conclude the final space group of a material (i.e. whether the system is centro-symmetric or and non-centro-symmetric).

Thus to find whether the crystal is centro-symmetric or and non-centro-symmetric we need to perform some useful physical properties measurements, like measurement of polarization. If we find our crystal is ferroelectric in nature then it must belong to a non-centro-symmetric space group.

(1) That is correct. But in X-ray diffraction some of the constituent atoms may have absorption edge close to the X-ray wavelength one uses in laboratory or synchrotron X-ray sources. If you suspect your structure can be non centrosymmetric then one should choose the wavelength accordingly.

(2) Try intensity statistics

(3) Measure physical properties that are sensitive to the presence or absence of centre of symmetry viz; ferroelectric polarization.

I would like to add following points to the on going discussions:

1) From the analysis of systematic absences (Please read any good crystallography book), one can guess the possible space group . Hence corresponding point group can be obtained. Then it can be checked that whether the point group has center of symmetry (i) or not .

For example: If your compound crystallizes in Monoclinic crystal system and systematic absences shows the presence of 21 screw axis and c-glide the possible space group would be P21/c , hence corresponding point group is '2/m' which has center of symmetry (i). therefore the system is centro symmetric. Similarly if in the same crystal system analysis of systematic absences shows the presence of C-center and and c-glide then the possible space group would be Cc and the corresponding point group be 'm'. Therefore no center of inversion presence, hence the  system is non-centrosymmetric.

2) One should be very careful in solving structure with non centrosymmetric space group. I would also like to recommend here to check data with E-statistics analysis (Acta Cryst. 1995, B51, 897) with Sheldrick's |E2 - 1| criterion and checking of final structure with ADDSYM analysis by PLATON (J. Appl. Cryst. 1987, 20, 264; J. Appl. Cryst. 1988, 21, 983). These analysis can be done with program suit WinGX. 

3) Experimental characterization of the compound (for example polarization measurement, SHG etc) would also be very helpful as also previously mentioned by various researchers.

There are only 230 unique combinations for three-dimensional symmetry, and these combinations are known as the 230 space groups. Each one of the 230 three-dimensional space groups is unique; but the choice of vectors that defines a unit cell for that symmetry is not unique. The space groups are numbered from 1 to 230 and are classified here according to the 7 crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Within each crystal system, the space groups can be ordered by Laue class, crystal class (e.g. 2 < m < 2/m) and, finally, lattice centring (e.g. P < A,B,C < F < I ).

Centrosymmetric: A point group which contains an inversion center as one of its symmetry elements is centrosymmetric. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Crystals with an inversion center cannot display certain properties, such as the electric polarization, piezoelectric effect.The following space groups have inversion symmetry: the triclinic space group No. 2, the monoclinic space group No. 10-15, the orthorhombic space group No. 47-74, the tetragonal space group No. 83-88 and 123-142, the trigonal space group No. 147, 148 and 162-167, the hexagonal space group No. 175, 176 and 191-194, the cubic space group No. 200-206 and 221-230.

Non-centrosymmetric: Non-centrosymmetric point groups lacks an inversion center and are divided into chiral and polar types.

Chiral: Some space groups have no symmetry element that can change the handedness of an object; these are termed as chiral.

Polar: A polar point group is whose symmetry operations leave more than one common point unmoved. A polar point group has no unique origin because each of those unmoved points can be chosen as one. The unmoved points collectively make a unique anisotropic axis. Polar point groups in crystal include 1, 2, 3, 4, 6, m, mm2, 3m, 4mm, and 6mm.

Note: Point groups 1, 2, 3, 4, 6 are chiral as well as polar.

Experimentally one should looking for electric polarization or second harmonic generation signal.

Dear Suman,

I think you got the lot of knowledge about the space group property by means of experiments like polarization ans SHG and so on.

Focusing on your question that i guess you wish to know that how to distinguish these two by just symmetry coordinate and the answer is simple. Just look at the general coordinate for space group (x y z), if there exist an (-x -y -z) then the space group is centrosymmetric. for example, P 1 non centrosymmetric and P -1 is a centrosymmetric.

Regards

A. Ahad

In wingx suite, there is a space group tool, where you can see whether the given space group belongs to centrosymmetric or non-centrosymmetric

If you know, how to find out the point group from the various space group. Then you only need to remember 11 Laue groups. Every Laue group gives the different centrosymmetric space group (must have an inversion symmetry) in its crystal system. Kindly go through this link.

http://pd.chem.ucl.ac.uk/pdnn/symm3/allsgp.htm.

Also, you can get an idea about whether a given space group is chiral or not. Please see attached .png file.