G is the group 32gp7
The Hall-Senior number of this group is 47.
G has 2 minimal generators, rank 3 and exponent 8. The centre has rank 1.
There are 2 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3.
This cohomology ring calculation is complete.
Ring structure | Completion information | Koszul information | Restriction information | Poincaré series
The cohomology ring has 8 generators:
There are 18 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 8. The cohomology ring approximation is stable from degree 8 onwards, and Benson's tests detect stability from degree 8 onwards.
This cohomology ring has dimension 3 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, h2, h3) is as follows.
A basis for AnnR/(h1, h2)(h3) is as follows.
A basis for AnnR/(h1)(h2) is as follows.
(1 + 2t + t2 + t3 + 2t4 + t5) / (1 - t2)2 (1 - t4)