David J. Green

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Cohomology of group number 573 of order 128

About the group

Ring generators

Ring relations

Completion information

Restriction maps

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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 1.
• 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 1.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t7  −  2·t6  +  t3  −  t  −  1

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

• The a-invariants are -∞,-4,-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 12 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. b_2_3, an element of degree 2
5. b_2_4, an element of degree 2
6. a_4_3, a nilpotent element of degree 4
7. b_5_7, an element of degree 5
8. a_6_4, a nilpotent element of degree 6
9. a_6_6, a nilpotent element of degree 6
10. a_7_10, a nilpotent element of degree 7
11. a_8_9, a nilpotent element of degree 8
12. c_8_15, a Duflot regular element of degree 8

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Ring relations

There are 42 minimal relations of maximal degree 16:

1. a_1_0²
2. a_1_1² + a_1_0·a_1_2
3. a_1_2² + a_1_1² + a_1_0·a_1_1
4. b_2_4·a_1_0 + b_2_3·a_1_2
5. b_2_3·b_2_4·a_1_2 + b_2_3²·a_1_2
6. a_4_3·a_1_0
7. a_4_3·a_1_2
8. a_1_0·b_5_7
9. a_1_1·a_1_2·b_5_7 + a_6_4·a_1_1 + b_2_3·a_4_3·a_1_1
10. a_6_4·a_1_0 + b_2_3·a_4_3·a_1_1
11. a_1_1·a_1_2·b_5_7 + a_6_4·a_1_2 + b_2_4·a_4_3·a_1_1
12. a_1_1·a_1_2·b_5_7 + a_6_6·a_1_1 + b_2_3·a_4_3·a_1_1
13. a_1_1·a_1_2·b_5_7 + a_6_6·a_1_0 + b_2_4·a_4_3·a_1_1
14. a_1_1·a_1_2·b_5_7 + a_6_6·a_1_2 + b_2_4·a_4_3·a_1_1
15. a_4_3²
16. b_2_4·a_1_2·b_5_7 + b_2_4²·a_4_3 + b_2_3·a_1_1·b_5_7 + b_2_3·b_2_4·a_4_3
17. b_2_4·a_6_4 + b_2_4²·a_4_3 + b_2_3·a_1_1·b_5_7 + b_2_3·a_6_6 + b_2_3²·a_4_3
18. b_2_4·a_6_6 + b_2_4·a_6_4 + a_1_1·a_7_10
19. a_1_0·a_7_10
20. a_1_2·a_7_10
21. a_4_3·b_5_7 + b_2_3²·b_2_4²·a_1_1 + b_2_3³·b_2_4·a_1_1 + b_2_3²·a_4_3·a_1_1
22. a_8_9·a_1_0 + b_2_3²·a_4_3·a_1_1
23. a_8_9·a_1_2 + b_2_3·b_2_4·a_4_3·a_1_1
24. b_5_7² + b_2_3·b_2_4⁴ + b_2_3³·b_2_4²
25. a_4_3·a_6_4
26. a_4_3·a_6_6
27. a_6_4·b_5_7 + b_2_3³·b_2_4²·a_1_1 + b_2_3⁴·b_2_4·a_1_1 + a_4_3·a_7_10
28. a_6_4·b_5_7 + b_2_3³·b_2_4²·a_1_1 + b_2_3⁴·b_2_4·a_1_1 + b_2_3·a_8_9·a_1_1
       + b_2_3³·a_4_3·a_1_1
29. a_6_6·b_5_7 + b_2_3³·b_2_4²·a_1_1 + b_2_3⁴·b_2_4·a_1_1 + b_2_4·a_8_9·a_1_1
       + b_2_4³·a_4_3·a_1_1 + b_2_3²·b_2_4·a_4_3·a_1_1 + b_2_3³·a_4_3·a_1_1
       + c_8_15·a_1_0·a_1_1·a_1_2
30. a_6_4²
31. a_6_6²
32. a_6_4·a_6_6
33. a_4_3·a_8_9 + b_2_3²·a_1_1·a_7_10
34. b_5_7·a_7_10 + b_2_4²·a_8_9 + b_2_4⁴·a_4_3 + b_2_3·b_2_4·a_8_9
       + b_2_3·b_2_4²·a_1_1·b_5_7 + b_2_3²·b_2_4·a_1_1·b_5_7 + b_2_3³·b_2_4·a_4_3
       + b_2_4²·a_1_1·a_7_10 + b_2_3·b_2_4·a_1_1·a_7_10
35. a_6_6·a_7_10 + a_6_4·a_7_10
36. a_6_6·a_7_10 + b_2_3²·a_8_9·a_1_1 + b_2_3⁴·a_4_3·a_1_1
37. a_8_9·b_5_7 + b_2_3·b_2_4²·a_7_10 + b_2_3²·b_2_4·a_7_10 + b_2_3⁴·b_2_4²·a_1_1
       + b_2_3⁵·b_2_4·a_1_1 + b_2_4²·a_8_9·a_1_1 + b_2_4⁴·a_4_3·a_1_1
       + b_2_3·b_2_4·a_8_9·a_1_1 + b_2_3³·b_2_4·a_4_3·a_1_1
38. a_7_10²
39. a_6_6·a_8_9 + b_2_3³·a_1_1·a_7_10
40. a_6_4·a_8_9 + b_2_3³·a_1_1·a_7_10
41. a_8_9·a_7_10 + b_2_3³·a_8_9·a_1_1 + b_2_3·a_4_3·c_8_15·a_1_1
42. a_8_9²

About the group

Ring generators

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 16.
• 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_8_15, a Duflot regular element of degree 8
  2. b_2_4² + b_2_3·b_2_4 + b_2_3², an element of degree 4
  3. b_2_4, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 11].
• 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 1

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. b_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. b_5_7 → 0, an element of degree 5
8. a_6_4 → 0, an element of degree 6
9. a_6_6 → 0, an element of degree 6
10. a_7_10 → 0, an element of degree 7
11. a_8_9 → 0, an element of degree 8
12. c_8_15 → 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. b_2_3 → c_1_1², an element of degree 2
5. b_2_4 → c_1_2², an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. b_5_7 → c_1_1·c_1_2⁴ + c_1_1³·c_1_2², an element of degree 5
8. a_6_4 → 0, an element of degree 6
9. a_6_6 → 0, an element of degree 6
10. a_7_10 → 0, an element of degree 7
11. a_8_9 → 0, an element of degree 8
12. c_8_15 → c_1_1⁴·c_1_2⁴ + c_1_1⁶·c_1_2² + c_1_0²·c_1_1²·c_1_2⁴
       + c_1_0²·c_1_1⁴·c_1_2² + c_1_0⁴·c_1_2⁴ + c_1_0⁴·c_1_1²·c_1_2²
       + c_1_0⁴·c_1_1⁴ + 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