David J. Green

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Cohomology of group number 878 of order 128

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General information on the group

• The group has 3 minimal generators and exponent 16.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 2.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

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

  t2  −  t  +  1

  (t  −  1)4 · (t2  +  1)

• 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 9 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. c_1_2, a Duflot regular element of degree 1
4. b_2_4, an element of degree 2
5. b_2_5, an element of degree 2
6. b_3_9, an element of degree 3
7. b_3_10, an element of degree 3
8. b_4_17, an element of degree 4
9. c_4_18, a Duflot regular element of degree 4

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Ring relations

There are 18 minimal relations of maximal degree 8:

1. a_1_0²
2. a_1_0·b_1_1
3. b_2_4·a_1_0
4. b_2_5·a_1_0
5. b_2_5·b_1_1² + b_2_4²
6. a_1_0·b_3_9
7. b_1_1·b_3_9 + b_2_4·b_2_5
8. a_1_0·b_3_10
9. b_2_5²·b_1_1 + b_2_4·b_3_9
10. b_4_17·a_1_0
11. b_4_17·b_1_1 + b_2_5²·b_1_1 + b_2_4·b_3_10 + b_2_4·b_2_5·b_1_1
12. b_3_9² + b_2_5³
13. b_3_10² + b_2_5·b_1_1·b_3_10 + c_4_18·b_1_1²
14. b_3_9·b_3_10 + b_2_5·b_4_17 + b_2_5³ + b_2_4·b_2_5²
15. b_2_5·b_1_1·b_3_10 + b_2_4·b_4_17 + b_2_4·b_2_5² + b_2_4²·b_2_5
16. b_4_17·b_3_10 + b_2_5²·b_3_10 + b_2_4·c_4_18·b_1_1
17. b_4_17·b_3_9 + b_2_5²·b_3_10 + b_2_5²·b_3_9 + b_2_4·b_2_5·b_3_9
18. b_4_17² + b_2_5⁴ + b_2_4·b_2_5·b_4_17 + b_2_4·b_2_5³ + b_2_4²·c_4_18

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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_1_2, a Duflot regular element of degree 1
  2. c_4_18, a Duflot regular element of degree 4
  3. b_1_1² + b_2_5 + b_2_4, an element of degree 2
  4. b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 1, 3, 5].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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. c_1_2 → c_1_0, an element of degree 1
4. b_2_4 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_3_9 → 0, an element of degree 3
7. b_3_10 → 0, an element of degree 3
8. b_4_17 → 0, an element of degree 4
9. c_4_18 → c_1_1⁴, 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. c_1_2 → c_1_0, an element of degree 1
4. b_2_4 → c_1_2·c_1_3, an element of degree 2
5. b_2_5 → c_1_3², an element of degree 2
6. b_3_9 → c_1_3³, an element of degree 3
7. b_3_10 → c_1_2²·c_1_3 + c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
8. b_4_17 → c_1_3⁴ + c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_2·c_1_3, an element of degree 4
9. c_4_18 → c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1·c_1_2·c_1_3² + c_1_1²·c_1_3²
      + c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

About the group

Ring generators

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Completion information

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Back to groups of order 128