David J. Green

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Cohomology of group number 216 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 3 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) · (t3  +  t2  +  1)

  (t  −  1)3 · (t2  +  1)2

• 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 4:

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_2, a nilpotent element of degree 2
5. a_3_4, a nilpotent element of degree 3
6. a_3_2, a nilpotent element of degree 3
7. b_3_6, an element of degree 3
8. a_4_9, a nilpotent element of degree 4
9. c_4_11, a Duflot regular element of degree 4
10. c_4_12, a Duflot regular element of degree 4

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Ring relations

There are 27 minimal relations of maximal degree 8:

1. a_1_0²
2. a_1_0·b_1_1
3. a_1_0·b_1_2²
4. b_1_1·b_1_2² + b_1_1²·b_1_2
5. a_2_2·a_1_0
6. a_2_2²
7. a_1_0·a_3_4
8. b_1_1·a_3_4 + a_2_2·b_1_2² + a_2_2·b_1_1·b_1_2
9. a_2_2·b_1_2² + a_2_2·b_1_1·b_1_2 + a_1_0·a_3_2
10. b_1_1·a_3_2
11. a_1_0·b_3_6
12. a_2_2·a_3_2 + a_2_2·a_3_4
13. a_1_0·b_1_2·a_3_2 + a_2_2·a_3_4
14. b_1_2²·b_3_6 + b_1_2⁵ + b_1_1·b_1_2·b_3_6 + b_1_1⁴·b_1_2 + b_1_2²·a_3_2
15. a_4_9·a_1_0 + a_2_2·a_3_4
16. a_4_9·b_1_1 + a_2_2·b_3_6 + a_2_2·b_1_1²·b_1_2 + a_2_2·a_3_4
17. a_3_4²
18. a_3_2²
19. a_3_2·b_3_6 + b_1_2³·a_3_2
20. a_3_4·b_3_6 + b_1_2³·a_3_4 + a_3_4·a_3_2
21. b_3_6² + b_1_2⁶ + a_2_2·b_1_1·b_3_6 + a_2_2·b_1_1³·b_1_2 + c_4_11·b_1_1²
22. a_2_2·a_4_9
23. a_4_9·b_1_2² + a_2_2·b_1_2·b_3_6 + a_2_2·b_1_1³·b_1_2 + a_3_4·a_3_2
24. a_4_9·a_3_2
25. a_4_9·a_3_4
26. a_4_9·b_3_6 + a_2_2·b_1_1·b_1_2·b_3_6 + a_2_2·b_1_1⁴·b_1_2 + b_1_2·a_3_4·a_3_2
       + a_2_2·c_4_11·b_1_1
27. a_4_9²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_11, a Duflot regular element of degree 4
  2. c_4_12, a Duflot regular element of degree 4
  3. b_1_2² + b_1_1·b_1_2 + 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_2 → 0, an element of degree 2
5. a_3_4 → 0, an element of degree 3
6. a_3_2 → 0, an element of degree 3
7. b_3_6 → 0, an element of degree 3
8. a_4_9 → 0, an element of degree 4
9. c_4_11 → c_1_0⁴, an element of degree 4
10. c_4_12 → c_1_1⁴, an element of degree 4

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_2 → 0, an element of degree 2
5. a_3_4 → 0, an element of degree 3
6. a_3_2 → 0, an element of degree 3
7. b_3_6 → c_1_0·c_1_2² + c_1_0²·c_1_2, an element of degree 3
8. a_4_9 → 0, an element of degree 4
9. c_4_11 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
10. c_4_12 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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_2 → 0, an element of degree 2
5. a_3_4 → 0, an element of degree 3
6. a_3_2 → 0, an element of degree 3
7. b_3_6 → c_1_2³, an element of degree 3
8. a_4_9 → 0, an element of degree 4
9. c_4_11 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
10. c_4_12 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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 → c_1_2, an element of degree 1
4. a_2_2 → 0, an element of degree 2
5. a_3_4 → 0, an element of degree 3
6. a_3_2 → 0, an element of degree 3
7. b_3_6 → c_1_0·c_1_2² + c_1_0²·c_1_2, an element of degree 3
8. a_4_9 → 0, an element of degree 4
9. c_4_11 → c_1_2⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
10. c_4_12 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128