David J. Green

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Cohomology of group number 488 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 3.
• Its center has rank 2.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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  +  t2  +  1)

  (t  −  1)3 · (t2  +  1) · (t4  +  1)

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

About the group

Ring generators

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Ring generators

The cohomology ring has 11 minimal generators of maximal degree 8:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_1_2, a nilpotent element of degree 1
4. a_2_3, a nilpotent element of degree 2
5. b_2_4, an 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. a_5_16, a nilpotent element of degree 5
9. b_5_18, an element of degree 5
10. a_6_21, a nilpotent element of degree 6
11. c_8_37, a Duflot regular element of degree 8

About the group

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Ring relations

There are 28 minimal relations of maximal degree 12:

1. a_1_1²
2. a_1_0·a_1_1
3. a_1_2² + a_1_0²
4. a_2_3·a_1_1
5. b_2_4·a_1_1 + a_1_0³
6. a_2_3² + a_2_3·a_1_0²
7. a_1_1·a_3_9 + a_2_3·a_1_0²
8. a_2_3·b_2_4 + a_1_0·a_3_9
9. b_2_4·a_1_0³
10. a_2_3·a_3_9 + a_1_0²·a_3_9
11. a_3_9² + b_2_4·a_1_0·a_3_9 + b_2_4²·a_1_0² + a_1_0³·a_3_9
12. a_1_1·a_5_16 + a_1_0³·a_3_9 + a_2_3·c_2_5·a_1_0²
13. a_1_0·a_5_16 + b_2_4²·a_1_0² + c_2_5·a_1_0·a_3_9
14. a_1_1·b_5_18 + a_2_3·c_2_5·a_1_0²
15. a_2_3·a_5_16 + b_2_4·a_1_0²·a_3_9 + c_2_5·a_1_0²·a_3_9
16. b_2_4·a_5_16 + b_2_4³·a_1_0 + a_1_0²·b_5_18 + b_2_4·c_2_5·a_3_9 + c_2_5·a_1_0²·a_3_9
17. a_6_21·a_1_1
18. a_2_3·b_5_18 + a_6_21·a_1_0 + b_2_4·a_1_0²·a_3_9 + a_2_3·c_2_5²·a_1_0
       + c_2_5²·a_1_0³
19. a_3_9·a_5_16 + b_2_4²·a_1_0·a_3_9 + a_6_21·a_1_0² + b_2_4·c_2_5·a_1_0·a_3_9
       + b_2_4²·c_2_5·a_1_0² + a_2_3·c_2_5²·a_1_0²
20. a_3_9·b_5_18 + b_2_4·a_6_21 + b_2_4²·a_1_0·a_3_9 + b_2_4³·a_1_0²
       + b_2_4²·c_2_5·a_1_0² + c_2_5·a_1_0³·a_3_9 + c_2_5²·a_1_0·a_3_9
       + b_2_4·c_2_5²·a_1_0²
21. a_3_9·a_5_16 + b_2_4²·a_1_0·a_3_9 + a_2_3·a_6_21 + b_2_4·c_2_5·a_1_0·a_3_9
       + b_2_4²·c_2_5·a_1_0²
22. a_6_21·a_3_9 + b_2_4·a_1_0²·b_5_18 + b_2_4·a_6_21·a_1_0 + b_2_4²·a_1_0²·a_3_9
       + b_2_4·c_2_5·a_1_0²·a_3_9 + c_2_5²·a_1_0²·a_3_9
23. a_5_16² + b_2_4⁴·a_1_0² + b_2_4·c_2_5²·a_1_0·a_3_9 + b_2_4²·c_2_5²·a_1_0²
       + c_2_5²·a_1_0³·a_3_9
24. a_5_16·b_5_18 + b_2_4²·a_1_0·b_5_18 + b_2_4⁴·a_1_0² + b_2_4·c_2_5·a_6_21
       + b_2_4²·c_2_5·a_1_0·a_3_9 + b_2_4³·c_2_5·a_1_0² + c_2_5·a_6_21·a_1_0²
       + b_2_4²·c_2_5²·a_1_0² + c_2_5³·a_1_0·a_3_9 + b_2_4·c_2_5³·a_1_0²
       + a_2_3·c_2_5³·a_1_0²
25. b_5_18² + b_2_4⁵ + b_2_4²·a_1_0·b_5_18 + b_2_4⁴·a_1_0² + b_2_4·a_6_21·a_1_0²
       + c_8_37·a_1_0² + b_2_4²·c_2_5·a_1_0·a_3_9 + b_2_4³·c_2_5·a_1_0²
       + b_2_4·c_2_5²·a_1_0·a_3_9 + b_2_4²·c_2_5²·a_1_0² + c_2_5²·a_1_0³·a_3_9
26. a_6_21·a_5_16 + b_2_4²·a_6_21·a_1_0 + b_2_4³·a_1_0²·a_3_9
       + b_2_4·c_2_5·a_1_0²·b_5_18 + b_2_4·c_2_5·a_6_21·a_1_0 + b_2_4²·c_2_5·a_1_0²·a_3_9
       + b_2_4·c_2_5²·a_1_0²·a_3_9 + c_2_5³·a_1_0²·a_3_9
27. a_6_21·b_5_18 + b_2_4⁴·a_3_9 + b_2_4²·a_1_0²·b_5_18 + b_2_4³·a_1_0²·a_3_9
       + b_2_4·c_2_5·a_1_0²·b_5_18 + a_2_3·c_8_37·a_1_0 + c_2_5²·a_1_0²·b_5_18
       + c_2_5²·a_6_21·a_1_0 + b_2_4·c_2_5²·a_1_0²·a_3_9 + a_2_3·c_2_5⁴·a_1_0
       + c_2_5⁴·a_1_0³
28. a_6_21² + b_2_4⁴·a_1_0·a_3_9 + b_2_4⁵·a_1_0² + b_2_4²·a_6_21·a_1_0²
       + a_2_3·c_8_37·a_1_0² + a_2_3·c_2_5⁴·a_1_0²

About the group

Ring generators

Ring relations

Completion information

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Back to groups of order 128

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_2_5, a Duflot regular element of degree 2
  2. c_8_37, a Duflot regular element of degree 8
  3. b_2_4, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 9].
• 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. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. a_2_3 → 0, an element of degree 2
5. b_2_4 → 0, 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. a_5_16 → 0, an element of degree 5
9. b_5_18 → 0, an element of degree 5
10. a_6_21 → 0, an element of degree 6
11. c_8_37 → c_1_1⁸ + c_1_0⁸, an element of degree 8

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

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. a_2_3 → 0, an element of degree 2
5. b_2_4 → c_1_2², 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. a_5_16 → 0, an element of degree 5
9. b_5_18 → c_1_2⁵, an element of degree 5
10. a_6_21 → 0, an element of degree 6
11. c_8_37 → c_1_2⁸ + c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0²·c_1_2⁶ + c_1_0⁴·c_1_2⁴ + c_1_0⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128