G = Q64 is Quaternion group of order 64
The Hall-Senior number of this group is 267.
G has 2 minimal generators, rank 1 and exponent 32. The centre has rank 1.
There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 1.
This cohomology ring calculation is complete.
Ring structure | Completion information | Koszul information | Restriction information | Poincaré series
The cohomology ring has 3 generators:
There are 2 minimal relations:
A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relation:
This cohomology ring was obtained from a calculation out to degree 12. The cohomology ring approximation is stable from degree 4 onwards, and Benson's tests detect stability from degree 4 onwards.
This cohomology ring has dimension 1 and depth 1. Here is a homogeneous system of parameters:
The first term h1 forms a regular sequence of maximum length.
The first term h1 forms a complete Duflot regular sequence. That is, its restriction to the greatest central elementary abelian subgroup forms a regular sequence of maximal length.
Data for Benson's test:
A basis for R/(h1) is as follows.
(1 + 2t + 2t2 + t3) / (1 - t4)