David J. Green

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Cohomology of group number 530 of order 128

About the group

Ring generators

Ring relations

Completion information

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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 4.
• Its center has rank 3.
• 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 3.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t2  −  t  −  1)

  (t  +  1)2 · (t  −  1)4

• The a-invariants are -∞,-∞,-∞,-4,-4. 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 9 minimal generators of maximal degree 4:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. b_1_2, an element of degree 1
4. a_2_3, a nilpotent element of degree 2
5. c_2_4, a Duflot regular element of degree 2
6. c_2_5, a Duflot regular element of degree 2
7. a_3_9, a nilpotent element of degree 3
8. b_3_10, an element of degree 3
9. c_4_16, a Duflot regular element of degree 4

About the group

Ring generators

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Ring relations

There are 14 minimal relations of maximal degree 6:

1. a_1_0²
2. a_1_1² + a_1_0·a_1_1
3. a_1_0·b_1_2
4. a_1_1·b_1_2²
5. a_2_3·a_1_0
6. a_2_3²
7. b_1_2·a_3_9
8. a_1_0·a_3_9 + a_2_3·a_1_1·b_1_2
9. a_1_1·b_3_10 + a_1_1·a_3_9
10. a_1_0·b_3_10 + a_1_1·a_3_9
11. a_2_3·a_3_9
12. a_3_9²
13. b_3_10² + a_3_9·b_3_10 + c_2_5·b_1_2⁴
14. a_3_9·b_3_10 + c_4_16·a_1_0·a_1_1

About the group

Ring generators

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Completion information

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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_4, a Duflot regular element of degree 2
  2. c_2_5, a Duflot regular element of degree 2
  3. c_4_16, a Duflot regular element of degree 4
  4. b_1_2², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 6].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. a_2_3 → 0, an element of degree 2
5. c_2_4 → c_1_1², an element of degree 2
6. c_2_5 → c_1_0², an element of degree 2
7. a_3_9 → 0, an element of degree 3
8. b_3_10 → 0, an element of degree 3
9. c_4_16 → c_1_2⁴, 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. a_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. a_2_3 → 0, an element of degree 2
5. c_2_4 → c_1_1·c_1_3 + c_1_1², an element of degree 2
6. c_2_5 → c_1_0², an element of degree 2
7. a_3_9 → 0, an element of degree 3
8. b_3_10 → c_1_0·c_1_3², an element of degree 3
9. c_4_16 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_0²·c_1_3², an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128