Small group number 257 of order 64

G is the group 64gp257

The Hall-Senior number of this group is 241.

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

The 15 maximal subgroups are: 32gp38, 32gp39 (3x), 32gp42 (3x), 32gp43 (6x), E32+ (2x).

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 8. 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 32gp49

Restriction to maximal subgroup number 2, which is 32gp49

Restriction to maximal subgroup number 3, which is 32gp38

Restriction to maximal subgroup number 4, which is 32gp39

Restriction to maximal subgroup number 5, which is 32gp43

Restriction to maximal subgroup number 6, which is 32gp43

Restriction to maximal subgroup number 7, which is 32gp42

Restriction to maximal subgroup number 8, which is 32gp39

Restriction to maximal subgroup number 9, which is 32gp43

Restriction to maximal subgroup number 10, which is 32gp43

Restriction to maximal subgroup number 11, which is 32gp42

Restriction to maximal subgroup number 12, which is 32gp42

Restriction to maximal subgroup number 13, which is 32gp43

Restriction to maximal subgroup number 14, which is 32gp43

Restriction to maximal subgroup number 15, which is 32gp39

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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