Small group number 49 of order 32

G = E32+ is Extraspecial 2-group of order 32 and type +

The Hall-Senior number of this group is 42.

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

The 15 maximal subgroups are: D8xC2 (9x), 16gp13 (6x).

There are 6 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 3, 3, 3, 3.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 5 generators:

There are 2 minimal relations:

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

Essential ideal: Zero ideal

Nilradical: Zero ideal

Completion information

This cohomology ring was obtained from a calculation out to degree 12. The cohomology ring approximation is stable from degree 4 onwards, and Carlson's tests detect stability from degree 8 onwards.

This cohomology ring has dimension 3 and depth 3. Here is a homogeneous system of parameters:

The first 3 terms h1, h2, h3 form 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.

The ideal of essential classes is the zero ideal. The essential ideal squares to zero.

Koszul information

A basis for R/(h1, h2, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 8.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 16gp11

Restriction to maximal subgroup number 2, which is 16gp11

Restriction to maximal subgroup number 3, which is 16gp11

Restriction to maximal subgroup number 4, which is 16gp11

Restriction to maximal subgroup number 5, which is 16gp11

Restriction to maximal subgroup number 6, which is 16gp13

Restriction to maximal subgroup number 7, which is 16gp13

Restriction to maximal subgroup number 8, which is 16gp11

Restriction to maximal subgroup number 9, which is 16gp13

Restriction to maximal subgroup number 10, which is 16gp11

Restriction to maximal subgroup number 11, which is 16gp13

Restriction to maximal subgroup number 12, which is 16gp13

Restriction to maximal subgroup number 13, which is 16gp11

Restriction to maximal subgroup number 14, which is 16gp11

Restriction to maximal subgroup number 15, which is 16gp13

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V8

Restriction to maximal elementary abelian number 2, which is V8

Restriction to maximal elementary abelian number 3, which is V8

Restriction to maximal elementary abelian number 4, which is V8

Restriction to maximal elementary abelian number 5, which is V8

Restriction to maximal elementary abelian number 6, which is V8

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C2

Poincaré series

(1 + 4t + 7t2 + 7t3 + 4t4 + t5) / (1 - t2)2 (1 - t4)

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