David J. Green

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Cohomology of group number 1710 of order 128

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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 4.
• Its center has rank 2.
• It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3 and 4, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  1

  (t  −  1)4 · (t2  +  1)2

• The a-invariants are -∞,-∞,-4,-4,-4. 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 6:

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_1_3, an element of degree 1
5. a_3_10, a nilpotent element of degree 3
6. b_3_11, an element of degree 3
7. c_4_16, a Duflot regular element of degree 4
8. c_4_17, a Duflot regular element of degree 4
9. b_5_27, an element of degree 5
10. a_6_24, a nilpotent element of degree 6

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Ring relations

There are 21 minimal relations of maximal degree 12:

1. a_1_0²
2. b_1_1·b_1_2 + a_1_0·b_1_3 + a_1_0·b_1_1
3. a_1_0·b_1_3² + a_1_0·b_1_2·b_1_3 + a_1_0·b_1_1·b_1_3
4. b_1_1·b_1_3² + b_1_1²·b_1_3 + a_1_0·b_1_3² + a_1_0·b_1_2² + a_1_0·b_1_1²
5. a_1_0·a_3_10
6. b_1_1·a_3_10 + a_1_0·b_3_11
7. b_1_3²·b_3_11 + b_1_2·b_1_3·b_3_11 + b_1_1·b_1_3·b_3_11 + b_1_2²·a_3_10
      + a_1_0·b_1_3·b_3_11 + a_1_0·b_1_1·b_3_11
8. a_3_10²
9. a_3_10·b_3_11 + a_1_0·b_1_1·b_1_3·b_3_11 + a_1_0·b_1_1²·b_3_11 + c_4_16·a_1_0·b_1_1
10. b_3_11² + b_1_1²·b_1_3·b_3_11 + b_1_1³·b_3_11 + a_1_0·b_1_1²·b_3_11
       + c_4_16·b_1_1²
11. a_3_10·b_3_11 + a_1_0·b_5_27 + a_1_0·b_1_1·b_1_3·b_3_11 + a_1_0·b_1_1²·b_3_11
       + c_4_16·a_1_0·b_1_3 + c_4_16·a_1_0·b_1_2
12. b_3_11² + b_1_1·b_5_27 + b_1_1²·b_1_3·b_3_11 + b_1_1³·b_3_11 + a_3_10·b_3_11
       + a_1_0·b_1_1·b_1_3·b_3_11 + a_1_0·b_1_1²·b_3_11 + a_1_0·b_1_1⁴·b_1_3
       + a_1_0·b_1_1⁵ + c_4_16·b_1_1·b_1_3 + c_4_16·a_1_0·b_1_3
13. a_6_24·a_1_0
14. a_1_0·b_1_1⁵·b_1_3 + a_1_0·b_1_1⁶ + a_6_24·b_1_1 + c_4_16·a_1_0·b_1_1·b_1_3
15. a_3_10·b_5_27 + b_1_3⁵·a_3_10 + b_1_2·b_1_3⁴·a_3_10 + b_1_2³·b_1_3²·a_3_10
       + b_1_2⁵·a_3_10 + a_1_0·b_1_1³·b_1_3·b_3_11 + a_6_24·b_1_3² + a_6_24·b_1_2·b_1_3
       + c_4_16·b_1_3·a_3_10 + c_4_16·b_1_2·a_3_10 + c_4_16·a_1_0·b_3_11
       + c_4_16·a_1_0·b_1_1²·b_1_3
16. b_3_11·b_5_27 + b_1_2⁴·b_1_3·b_3_11 + b_1_2⁵·b_3_11 + b_1_2²·b_1_3³·a_3_10
       + b_1_2⁵·a_3_10 + a_6_24·b_1_2² + c_4_16·b_1_3·b_3_11 + c_4_16·b_1_2·b_3_11
       + c_4_16·b_1_1·b_3_11 + c_4_16·a_1_0·b_1_2²·b_1_3 + c_4_16·a_1_0·b_1_1³
17. a_6_24·a_3_10
18. a_1_0·b_1_1⁴·b_1_3·b_3_11 + a_1_0·b_1_1⁵·b_3_11 + a_6_24·b_3_11
       + c_4_16·a_1_0·b_1_3·b_3_11
19. b_5_27² + b_1_2·b_1_3⁴·b_5_27 + b_1_2⁴·b_1_3·b_5_27 + b_1_2⁶·b_1_3·b_3_11
       + b_1_3⁷·a_3_10 + b_1_2·b_1_3⁶·a_3_10 + b_1_2⁴·b_1_3³·a_3_10
       + b_1_2⁵·b_1_3²·a_3_10 + a_1_0·b_1_1⁶·b_3_11 + a_6_24·b_1_1·b_3_11
       + c_4_17·b_1_2²·b_1_3⁴ + c_4_17·b_1_2⁴·b_1_3² + c_4_16·b_1_3⁶
       + c_4_16·b_1_2·b_1_3⁵ + c_4_16·b_1_2⁴·b_1_3² + c_4_16·b_1_2⁵·b_1_3
       + c_4_16·b_1_2⁶ + c_4_16·b_1_1⁵·b_1_3 + c_4_16·a_1_0·b_1_1·b_1_3·b_3_11
       + c_4_16·a_1_0·b_1_1⁴·b_1_3 + c_4_16·a_1_0·b_1_1⁵ + c_4_16²·b_1_3²
       + c_4_16²·b_1_2² + c_4_16²·b_1_1²
20. b_1_2⁷·b_1_3·b_3_11 + b_1_2⁸·b_3_11 + b_1_3⁸·a_3_10 + b_1_2⁵·b_1_3³·a_3_10
       + b_1_2⁶·b_1_3²·a_3_10 + a_1_0·b_1_1⁷·b_3_11 + a_6_24·b_5_27 + a_6_24·b_1_3⁵
       + a_6_24·b_1_2³·b_1_3² + a_6_24·b_1_2⁴·b_1_3 + a_6_24·b_1_2⁵
       + a_6_24·b_1_1²·b_3_11 + c_4_16·b_1_2⁴·b_3_11 + c_4_17·b_1_2²·b_1_3²·a_3_10
       + c_4_17·b_1_2³·b_1_3·a_3_10 + c_4_16·b_1_3⁴·a_3_10 + c_4_16·b_1_2·b_1_3³·a_3_10
       + c_4_16·a_1_0·b_1_1⁶ + c_4_16·a_6_24·b_1_3 + c_4_16·a_6_24·b_1_2
       + c_4_16²·a_1_0·b_1_1·b_1_3
21. a_6_24²

About the group

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 12.
• 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_16, a Duflot regular element of degree 4
  2. c_4_17, a Duflot regular element of degree 4
  3. b_1_3² + b_1_2·b_1_3 + b_1_2² + b_1_1·b_1_3 + b_1_1², an element of degree 2
  4. b_1_3², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6, 8].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

Ring generators

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Back to groups of order 128

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_1_3 → 0, an element of degree 1
5. a_3_10 → 0, an element of degree 3
6. b_3_11 → 0, an element of degree 3
7. c_4_16 → c_1_0⁴, an element of degree 4
8. c_4_17 → c_1_1⁴ + c_1_0⁴, an element of degree 4
9. b_5_27 → 0, an element of degree 5
10. a_6_24 → 0, an element of degree 6

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. b_1_3 → 0, an element of degree 1
5. a_3_10 → 0, an element of degree 3
6. b_3_11 → c_1_0·c_1_2² + c_1_0²·c_1_2, an element of degree 3
7. c_4_16 → c_1_0·c_1_2³ + c_1_0⁴, an element of degree 4
8. c_4_17 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
9. b_5_27 → c_1_0·c_1_2⁴ + c_1_0⁴·c_1_2, an element of degree 5
10. a_6_24 → 0, an element of degree 6

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. b_1_3 → c_1_2, an element of degree 1
5. a_3_10 → 0, an element of degree 3
6. b_3_11 → c_1_0·c_1_2² + c_1_0²·c_1_2, an element of degree 3
7. c_4_16 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
8. c_4_17 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
9. b_5_27 → 0, an element of degree 5
10. a_6_24 → 0, an element of degree 6

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_1_3 → c_1_3, an element of degree 1
5. a_3_10 → 0, an element of degree 3
6. b_3_11 → 0, an element of degree 3
7. c_4_16 → 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
8. c_4_17 → c_1_1²·c_1_3² + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_3³
      + c_1_0·c_1_2²·c_1_3 + c_1_0·c_1_2³ + c_1_0⁴, an element of degree 4
9. b_5_27 → c_1_1²·c_1_2·c_1_3² + c_1_1²·c_1_2²·c_1_3 + c_1_0·c_1_3⁴ + c_1_0·c_1_2·c_1_3³
      + c_1_0·c_1_2²·c_1_3² + c_1_0·c_1_2³·c_1_3 + c_1_0·c_1_2⁴ + c_1_0²·c_1_2²·c_1_3
      + c_1_0⁴·c_1_3 + c_1_0⁴·c_1_2, an element of degree 5
10. a_6_24 → 0, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128