David J. Green

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Cohomology of group number 2087 of order 128

About the group

Ring generators

Ring relations

Completion information

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General information on the group

• The group has 4 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• It has 4 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) · (t5  +  2·t4  +  2·t2  +  2·t  +  1)

  (t  +  1) · (t  −  1)3 · (t2  +  1)2

• The a-invariants are -∞,-∞,-5,-3. They were obtained using the filter regular HSOP of the Benson test.

About the group

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 4:

1. b_1_0, an element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an element of degree 1
4. b_1_3, an element of degree 1
5. a_4_5, a nilpotent element of degree 4
6. a_4_8, a nilpotent element of degree 4
7. b_4_9, an element of degree 4
8. b_4_11, an element of degree 4
9. c_4_12, a Duflot regular element of degree 4
10. c_4_13, a Duflot regular element of degree 4

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Ring relations

There are 23 minimal relations of maximal degree 8:

1. b_1_0·b_1_2
2. b_1_2² + b_1_1·b_1_3 + b_1_0·b_1_1 + b_1_0²
3. b_1_1·b_1_2·b_1_3 + b_1_1²·b_1_2 + b_1_0·b_1_3²
4. b_1_1·b_1_3² + b_1_1²·b_1_3 + b_1_0·b_1_1² + b_1_0²·b_1_1
5. a_4_5·b_1_0
6. a_4_8·b_1_2 + a_4_5·b_1_2
7. b_1_0·b_1_1⁴ + b_1_0⁵ + a_4_8·b_1_3 + a_4_5·b_1_3 + a_4_5·b_1_2 + a_4_5·b_1_1
8. b_4_9·b_1_3 + b_4_9·b_1_2 + a_4_5·b_1_3
9. b_4_9·b_1_0
10. b_1_0·b_1_1⁴ + b_1_0⁵ + b_4_9·b_1_2 + b_4_9·b_1_1 + a_4_5·b_1_2
11. b_4_11·b_1_2 + b_4_9·b_1_2 + a_4_5·b_1_2
12. b_4_11·b_1_3 + b_4_9·b_1_2 + a_4_8·b_1_0
13. b_4_11·b_1_1 + b_4_11·b_1_0 + b_4_9·b_1_2 + a_4_8·b_1_1 + a_4_5·b_1_2
14. a_4_5·b_1_2·b_1_3³ + a_4_5·b_1_1³·b_1_2 + a_4_5²
15. a_4_8·b_1_0·b_1_1³ + a_4_8·b_1_0⁴ + a_4_5·b_1_2·b_1_3³ + a_4_5·b_1_1³·b_1_2
       + a_4_8²
16. a_4_8² + a_4_5·a_4_8
17. b_1_1⁷·b_1_3 + b_1_1⁷·b_1_2 + b_1_0⁷·b_1_1 + b_1_0⁸ + a_4_8·b_4_9
       + a_4_5·b_1_2·b_1_3³ + a_4_5·b_1_1⁴ + c_4_13·b_1_1³·b_1_3 + c_4_13·b_1_1³·b_1_2
       + c_4_13·b_1_0·b_1_1³ + c_4_13·b_1_0⁴ + c_4_12·b_1_1³·b_1_3 + c_4_12·b_1_1³·b_1_2
       + c_4_12·b_1_0·b_1_1³ + c_4_12·b_1_0⁴
18. b_1_1⁷·b_1_3 + b_1_1⁷·b_1_2 + b_1_0⁷·b_1_1 + b_1_0⁸ + a_4_5·b_1_1³·b_1_2
       + a_4_5·b_1_1⁴ + a_4_5·b_4_9 + a_4_8² + c_4_13·b_1_1³·b_1_3 + c_4_13·b_1_1³·b_1_2
       + c_4_13·b_1_0·b_1_1³ + c_4_13·b_1_0⁴ + c_4_12·b_1_1³·b_1_3 + c_4_12·b_1_1³·b_1_2
       + c_4_12·b_1_0·b_1_1³ + c_4_12·b_1_0⁴
19. b_1_1⁷·b_1_3 + b_1_0⁷·b_1_1 + b_1_0⁸ + b_4_9² + a_4_5·b_1_1³·b_1_2 + a_4_5·b_1_1⁴
       + a_4_8² + c_4_13·b_1_1³·b_1_3 + c_4_13·b_1_0·b_1_1³ + c_4_13·b_1_0⁴
       + c_4_12·b_1_1³·b_1_3 + c_4_12·b_1_0·b_1_1³ + c_4_12·b_1_0⁴
20. b_1_1⁷·b_1_3 + b_1_1⁷·b_1_2 + b_1_0⁷·b_1_1 + b_1_0⁸ + a_4_8·b_4_11
       + a_4_5·b_1_1³·b_1_2 + a_4_5·b_1_1⁴ + c_4_13·b_1_1³·b_1_3 + c_4_13·b_1_1³·b_1_2
       + c_4_13·b_1_0·b_1_1³ + c_4_13·b_1_0³·b_1_1 + c_4_12·b_1_1³·b_1_3
       + c_4_12·b_1_1³·b_1_2 + c_4_12·b_1_0·b_1_1³ + c_4_12·b_1_0³·b_1_1
21. b_1_1⁷·b_1_3 + b_1_1⁷·b_1_2 + b_1_0⁷·b_1_1 + b_1_0⁸ + a_4_5·b_1_1³·b_1_2
       + a_4_5·b_1_1⁴ + a_4_5·b_4_11 + c_4_13·b_1_1³·b_1_3 + c_4_13·b_1_1³·b_1_2
       + c_4_13·b_1_0·b_1_1³ + c_4_13·b_1_0⁴ + c_4_12·b_1_1³·b_1_3 + c_4_12·b_1_1³·b_1_2
       + c_4_12·b_1_0·b_1_1³ + c_4_12·b_1_0⁴
22. b_1_1⁷·b_1_3 + b_1_0⁷·b_1_1 + b_1_0⁸ + b_4_11² + a_4_8·b_1_0⁴
       + a_4_5·b_1_1³·b_1_2 + a_4_5·b_1_1⁴ + c_4_13·b_1_1³·b_1_3 + c_4_13·b_1_0·b_1_1³
       + c_4_12·b_1_1³·b_1_3 + c_4_12·b_1_0·b_1_1³
23. b_1_1⁷·b_1_2 + b_4_9·b_4_11 + a_4_5·b_1_2·b_1_3³ + a_4_5·b_1_1³·b_1_2 + a_4_8²
       + c_4_13·b_1_1³·b_1_2 + c_4_12·b_1_1³·b_1_2

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

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_12, a Duflot regular element of degree 4
  2. c_4_13, a Duflot regular element of degree 4
  3. b_1_3² + b_1_2² + b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Restriction maps

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

1. b_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_1_3 → 0, an element of degree 1
5. a_4_5 → 0, an element of degree 4
6. a_4_8 → 0, an element of degree 4
7. b_4_9 → 0, an element of degree 4
8. b_4_11 → 0, an element of degree 4
9. c_4_12 → c_1_0⁴, an element of degree 4
10. c_4_13 → c_1_1⁴, an element of degree 4

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

1. b_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_1_3 → 0, an element of degree 1
5. a_4_5 → 0, an element of degree 4
6. a_4_8 → 0, an element of degree 4
7. b_4_9 → 0, an element of degree 4
8. b_4_11 → 0, an element of degree 4
9. c_4_12 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
10. c_4_13 → 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. b_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_1_3 → c_1_2, an element of degree 1
5. a_4_5 → 0, an element of degree 4
6. a_4_8 → 0, an element of degree 4
7. b_4_9 → 0, an element of degree 4
8. b_4_11 → 0, an element of degree 4
9. c_4_12 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
10. c_4_13 → 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. b_1_0 → c_1_2, 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_1_3 → 0, an element of degree 1
5. a_4_5 → 0, an element of degree 4
6. a_4_8 → 0, an element of degree 4
7. b_4_9 → 0, an element of degree 4
8. b_4_11 → c_1_1·c_1_2³ + c_1_1²·c_1_2² + c_1_0·c_1_2³ + c_1_0²·c_1_2², an element of degree 4
9. c_4_12 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
10. c_4_13 → 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. b_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_1_3 → c_1_2, an element of degree 1
5. a_4_5 → 0, an element of degree 4
6. a_4_8 → 0, an element of degree 4
7. b_4_9 → c_1_2⁴ + c_1_1·c_1_2³ + c_1_1²·c_1_2² + c_1_0·c_1_2³ + c_1_0²·c_1_2², an element of degree 4
8. b_4_11 → c_1_2⁴ + c_1_1·c_1_2³ + c_1_1²·c_1_2² + c_1_0·c_1_2³ + c_1_0²·c_1_2², an element of degree 4
9. c_4_12 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
10. c_4_13 → 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