David J. Green

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Cohomology of group number 522 of order 128

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General information on the group

• The group has 3 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 2.
• It has 5 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 4.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  t4  +  t3  +  3·t2  +  t  +  1

  (t  +  1)2 · (t  −  1)4 · (t2  +  1)

• The a-invariants are -∞,-∞,-∞,-∞,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 11 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. b_2_3, an element of degree 2
5. b_2_4, an element of degree 2
6. b_2_6, an element of degree 2
7. b_2_7, an element of degree 2
8. b_2_5, an element of degree 2
9. c_2_8, a Duflot regular element of degree 2
10. b_3_16, an element of degree 3
11. c_4_30, a Duflot regular element of degree 4

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Ring relations

There are 21 minimal relations of maximal degree 6:

1. a_1_0²
2. a_1_0·b_1_1
3. a_1_0·b_1_2
4. b_2_3·a_1_0
5. b_2_4·a_1_0
6. b_2_4·b_1_1 + b_2_3·b_1_2
7. b_2_7·b_1_2 + b_2_6·b_1_2 + b_2_7·a_1_0 + b_2_6·a_1_0
8. b_2_7·b_1_1 + b_2_6·b_1_2 + b_2_6·a_1_0
9. b_2_5·b_1_1 + b_2_6·b_1_2 + b_2_3·b_1_2 + b_2_6·a_1_0
10. b_2_3·b_1_2² + b_2_3·b_1_1·b_1_2 + b_2_3·b_2_4 + b_2_3² + c_2_8·b_1_1·b_1_2
       + c_2_8·b_1_1²
11. b_2_6·b_1_1·b_1_2 + b_2_3·b_1_1·b_1_2 + b_2_3² + c_2_8·b_1_1²
12. b_2_4·b_2_6 + b_2_3·b_2_7
13. b_2_5·b_1_2² + b_2_4² + c_2_8·b_1_2²
14. b_2_4·b_2_7 + b_2_3·b_1_2² + b_2_3·b_1_1·b_1_2 + b_2_3·b_2_5 + b_2_3²
       + c_2_8·b_1_1·b_1_2 + c_2_8·b_1_1²
15. b_2_7² + b_2_6·b_2_5 + b_2_4·b_2_7
16. b_1_2·b_3_16 + b_2_4·b_2_7 + b_2_4·b_2_6
17. b_2_4·b_2_7 + b_2_4·b_2_6 + a_1_0·b_3_16
18. b_1_1·b_3_16
19. b_2_3·b_3_16 + b_2_6·b_2_5·a_1_0 + b_2_6·b_2_7·a_1_0
20. b_2_4·b_3_16 + b_2_7·b_2_5·a_1_0 + b_2_6·b_2_5·a_1_0
21. b_3_16² + b_2_6·b_2_5² + b_2_6²·b_2_5 + b_2_3·b_2_6·b_2_7 + b_2_3²·b_2_6
       + b_2_5·a_1_0·b_3_16 + b_2_7·a_1_0·b_3_16 + b_2_6·a_1_0·b_3_16 + b_2_6·c_2_8·b_1_1²
       + b_2_3·c_2_8·b_1_1·b_1_2 + b_2_3²·c_2_8 + c_2_8²·b_1_1²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 6.
• 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_2_8, a Duflot regular element of degree 2
  2. c_4_30, a Duflot regular element of degree 4
  3. b_1_2² + b_1_1·b_1_2 + b_1_1² + b_2_5 + b_2_7 + b_2_6 + b_2_4, an element of degree 2
  4. b_3_16 + b_1_1·b_1_2² + b_1_1²·b_1_2 + b_2_5·b_1_2 + b_2_6·b_1_2 + b_2_6·b_1_1
        + b_2_4·b_1_2, an element of degree 3
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
• We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

About the group

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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. b_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_2_7 → 0, an element of degree 2
8. b_2_5 → 0, an element of degree 2
9. c_2_8 → c_1_1², an element of degree 2
10. b_3_16 → 0, an element of degree 3
11. c_4_30 → c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

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_3, an element of degree 1
4. b_2_3 → c_1_1·c_1_2, an element of degree 2
5. b_2_4 → c_1_1·c_1_3, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_2_7 → 0, an element of degree 2
8. b_2_5 → c_1_1·c_1_3, an element of degree 2
9. c_2_8 → c_1_1·c_1_3 + c_1_1², an element of degree 2
10. b_3_16 → 0, an element of degree 3
11. c_4_30 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
       + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

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. b_2_3 → c_1_1·c_1_2, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. b_2_6 → c_1_3² + c_1_2·c_1_3, an element of degree 2
7. b_2_7 → 0, an element of degree 2
8. b_2_5 → 0, an element of degree 2
9. c_2_8 → c_1_1², an element of degree 2
10. b_3_16 → 0, an element of degree 3
11. c_4_30 → c_1_3⁴ + c_1_2²·c_1_3² + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3
       + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

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. b_2_3 → 0, an element of degree 2
5. b_2_4 → c_1_1·c_1_2, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_2_7 → 0, an element of degree 2
8. b_2_5 → c_1_3² + c_1_2·c_1_3 + c_1_1·c_1_2, an element of degree 2
9. c_2_8 → c_1_3² + c_1_2·c_1_3 + c_1_1·c_1_2 + c_1_1², an element of degree 2
10. b_3_16 → 0, an element of degree 3
11. c_4_30 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
       + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

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. b_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. b_2_6 → c_1_2², an element of degree 2
7. b_2_7 → c_1_2·c_1_3, an element of degree 2
8. b_2_5 → c_1_3², an element of degree 2
9. c_2_8 → c_1_3² + c_1_1², an element of degree 2
10. b_3_16 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
11. c_4_30 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_1²·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_3²
       + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2²
       + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

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. b_2_3 → c_1_1·c_1_2, an element of degree 2
5. b_2_4 → c_1_1·c_1_2, an element of degree 2
6. b_2_6 → c_1_3² + c_1_2·c_1_3, an element of degree 2
7. b_2_7 → c_1_3² + c_1_2·c_1_3, an element of degree 2
8. b_2_5 → c_1_3² + c_1_2·c_1_3 + c_1_1·c_1_2, an element of degree 2
9. c_2_8 → c_1_3² + c_1_2·c_1_3 + c_1_1·c_1_2 + c_1_1², an element of degree 2
10. b_3_16 → 0, an element of degree 3
11. c_4_30 → c_1_1²·c_1_3² + c_1_1²·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3
       + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128