Small group number 515 of order 128

G is the group 128gp515

G has 3 minimal generators, rank 5 and exponent 8. The centre has rank 2.

There are 2 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 5, 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 18. The cohomology ring approximation is stable from degree 10 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