G is the group 32gp12
The Hall-Senior number of this group is 21.
G has 2 minimal generators, rank 2 and exponent 8. The centre has rank 2.
The 3 maximal subgroups are: Ab(4,4), Ab(8,2) (2x).
There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 2.
This cohomology ring calculation is complete.
Ring structure | Completion information | Koszul information | Restriction information | Poincaré series
The cohomology ring has 4 generators:
There are 2 minimal relations:
This minimal generating set constitutes a Gröbner basis for the relations ideal.
Essential ideal: There is one minimal generator:
Nilradical: There are 2 minimal generators:
This cohomology ring was obtained from a calculation out to degree 12. The cohomology ring approximation is stable from degree 2 onwards, and Carlson's tests detect stability from degree 4 onwards.
This cohomology ring has dimension 2 and depth 2. Here is a homogeneous system of parameters:
The first 2 terms h1, h2 form a regular sequence of maximum length.
The first 2 terms h1, h2 form a complete Duflot regular sequence. That is, their restrictions to the greatest central elementary abelian subgroup form a regular sequence of maximal length.
The ideal of essential classes is free of rank 1 as a module over the polynomial algebra on h1, h2. These free generators are:
A basis for R/(h1, h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 4.
(1 + 2t + t2) / (1 - t2)2