Small group number 48 of order 128

G is the group 128gp48

G has 2 minimal generators, rank 5 and exponent 8. The centre has rank 2.

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 5.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 17 generators:

There are 86 minimal relations:

This minimal generating set constitutes a Gröbner basis for the relations ideal.

Completion information

This cohomology ring was obtained from a calculation out to degree 13. The cohomology ring approximation is stable from degree 12 onwards, and Benson's tests detect stability from degree 13 onwards.

This cohomology ring has dimension 5 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.

Data for Benson's test:

Koszul information

A basis for R/(h1, h2, h3, h4, h5) is as follows.

A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.

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

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

Restriction information