David J. Green

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Cohomology of group number 227 of order 128

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General information on the group

• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  − 1

  (t  −  1)3 · (t2  +  1)

• The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 5:

1. a_1_0, a nilpotent element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an element of degree 1
4. a_2_4, a nilpotent element of degree 2
5. a_3_5, a nilpotent element of degree 3
6. a_3_6, a nilpotent element of degree 3
7. b_4_7, an element of degree 4
8. c_4_8, a Duflot regular element of degree 4
9. c_4_9, a Duflot regular element of degree 4
10. a_5_8, a nilpotent element of degree 5

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Ring relations

There are 27 minimal relations of maximal degree 10:

1. a_1_0²
2. a_1_0·b_1_1
3. b_1_1·b_1_2²
4. b_1_1²·b_1_2 + a_1_0·b_1_2²
5. a_2_4·a_1_0
6. a_2_4²
7. a_2_4·b_1_1·b_1_2 + a_1_0·a_3_5
8. b_1_1·a_3_5 + a_2_4·b_1_2²
9. a_2_4·b_1_2² + a_1_0·a_3_6
10. b_1_1·a_3_6
11. a_2_4·a_3_5
12. a_2_4·a_3_6
13. b_4_7·b_1_2 + b_1_2²·a_3_6
14. b_4_7·a_1_0
15. a_3_5²
16. a_3_6²
17. b_1_2·a_5_8 + a_3_5·a_3_6
18. a_1_0·a_5_8
19. b_1_1·a_5_8 + a_2_4·b_4_7
20. b_4_7·a_3_6
21. b_4_7·a_3_5 + b_1_2·a_3_5·a_3_6
22. a_2_4·a_5_8
23. b_4_7² + c_4_9·b_1_1⁴ + c_4_8·b_1_1⁴
24. a_3_6·a_5_8
25. a_3_5·a_5_8
26. b_4_7·a_5_8 + a_2_4·c_4_9·b_1_1³ + a_2_4·c_4_8·b_1_1³
27. a_5_8²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 10.
• The completion test was perfect: It applied in the last degree in which a generator or relation was found.
• The following is a filter regular homogeneous system of parameters:
  1. c_4_8, a Duflot regular element of degree 4
  2. c_4_9, a Duflot regular element of degree 4
  3. b_1_2² + b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. a_3_5 → 0, an element of degree 3
6. a_3_6 → 0, an element of degree 3
7. b_4_7 → 0, an element of degree 4
8. c_4_8 → c_1_1⁴ + c_1_0⁴, an element of degree 4
9. c_4_9 → c_1_1⁴, an element of degree 4
10. a_5_8 → 0, an element of degree 5

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. a_3_5 → 0, an element of degree 3
6. a_3_6 → 0, an element of degree 3
7. b_4_7 → c_1_0·c_1_2³ + c_1_0²·c_1_2², an element of degree 4
8. c_4_8 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
9. c_4_9 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
10. a_5_8 → 0, an element of degree 5

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. a_3_5 → 0, an element of degree 3
6. a_3_6 → 0, an element of degree 3
7. b_4_7 → 0, an element of degree 4
8. c_4_8 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
9. c_4_9 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
10. a_5_8 → 0, an element of degree 5

About the group

Ring generators

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Completion information

Restriction maps

Back to groups of order 128