Small group number 52 of order 128

G is the group 128gp52

G has 2 minimal generators, rank 4 and exponent 16. 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 27 generators:

There are 288 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 20. The cohomology ring approximation is stable from degree 20 onwards, and Benson's tests detect stability from degree 20 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