G = Q8 is Quaternion group of order 8
The Hall-Senior number of this group is 5.
G has 2 minimal generators, rank 1 and exponent 4. 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:
This minimal generating set constitutes a Gröbner basis for the relations ideal.
This cohomology ring was obtained from a calculation out to degree 4. 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)