# Noncentrosymmetric crystal symmetries; list of polar and chiral space groups

In the following webpages, I can find a list of polar or chiral point/space groups.

However, I can't figure out why the space group P-6m2 #187 is non-centrosymmetric?

Where can I also find the list of non-centrosymmetric space groups?

A space group is non-centrosymmetric if it doesn't contain centers of inversion.
To check for centers of inversion, you need to look at the full Hermann-Mauguin symbol.

If the symbol contains
(a) rotoinversion axes of odd order (-n with n odd)
and/or
(b) rotation/screw axes of even order perpendicular to planes of (glide) reflection (e.g. 2/m, 21/c, 41/a etc.),
then the space group has centers of inversion (i.e. is centrosymmetric),
else it is non-centrosymmetric.

Note:
Since centers of inversion are points, they are not associated with particular symmetry directions. Thus, they should be simultaneously present (or absent) in all symmetry directions of the full Hermann-Mauguin symbol.
The only exception from this rule is when "1" is used as a place holder in symmetry directions without other symmetry elements, as e.g. in space group P -3 1 2/m (#162). In this case, only -3 and 2/m betray the centers of inversion, while 1 doesn't because it is just a place holder.

Example:
P -6 m 2 is both the short and the full Hermann-Mauguin symbol for space group #187. It has neither odd rotoinversions nor even rotation axes perpendicular to planes of (glide) reflections, thus this space group is non-centrosymmetric.