Small group number 823 of order 128

G is the group 128gp823

G has 3 minimal generators, rank 4 and exponent 8. The centre has rank 1.

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 4.

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series

Ring structure

The cohomology ring has 20 generators:

There are 144 minimal relations:

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relations:

Completion information

This cohomology ring was obtained from a calculation out to degree 18. The cohomology ring approximation is stable from degree 18 onwards, and Benson's tests detect stability from degree 18 onwards.

This cohomology ring has dimension 4 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:

Koszul information

A basis for R/(h1, h2, h3, h4) is as follows.

A basis for AnnR/(h1, h2, h3)(h4) is as follows.

A basis for AnnR/(h1, h2)(h3) is as follows.

A basis for AnnR/(h1)(h2) is as follows.

Restriction information