David J. Green

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Cohomology of group number 51 of order 128

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Ring generators

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General information on the group

• The group has 2 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

  (t  +  1) · (t  −  1)3

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

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Ring generators

The cohomology ring has 14 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_2_1, a nilpotent element of degree 2
4. b_2_2, an element of degree 2
5. c_2_3, a Duflot regular element of degree 2
6. a_3_4, a nilpotent element of degree 3
7. a_3_5, a nilpotent element of degree 3
8. a_4_5, a nilpotent element of degree 4
9. a_5_9, a nilpotent element of degree 5
10. a_5_10, a nilpotent element of degree 5
11. a_6_9, a nilpotent element of degree 6
12. a_7_16, a nilpotent element of degree 7
13. a_7_19, a nilpotent element of degree 7
14. c_8_24, a Duflot regular element of degree 8

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Ring relations

There are 65 minimal relations of maximal degree 14:

1. a_1_0²
2. a_1_0·a_1_1
3. a_2_1·a_1_1
4. a_2_1·a_1_0
5. b_2_2·a_1_0 + a_1_1³
6. a_2_1²
7. a_1_0·a_3_4
8. a_1_1·a_3_5 + a_1_1·a_3_4
9. a_1_0·a_3_5
10. b_2_2·a_1_1³
11. a_2_1·a_3_4
12. a_2_1·a_3_5 + a_1_1²·a_3_4
13. b_2_2·a_3_4 + a_4_5·a_1_1
14. a_4_5·a_1_0 + a_1_1²·a_3_4
15. a_3_4²
16. a_3_5²
17. a_3_4·a_3_5
18. a_2_1·a_4_5
19. a_1_1·a_5_9 + a_4_5·a_1_1² + c_2_3²·a_1_1²
20. a_1_0·a_5_9
21. a_1_0·a_5_10
22. a_4_5·a_3_5
23. a_4_5·a_3_4 + a_4_5·a_1_1³
24. a_2_1·a_5_9
25. a_2_1·a_5_10 + a_4_5·a_1_1³
26. b_2_2·a_5_9 + b_2_2²·a_3_5 + a_1_1²·a_5_10 + b_2_2·c_2_3²·a_1_1
27. a_6_9·a_1_1 + c_2_3·a_1_1²·a_3_4
28. a_6_9·a_1_0 + a_4_5·a_1_1³
29. a_4_5² + b_2_2²·c_2_3·a_1_1²
30. a_3_5·a_5_9 + c_2_3²·a_1_1·a_3_4
31. a_3_4·a_5_9 + c_2_3²·a_1_1·a_3_4
32. a_2_1·b_2_2³ + a_3_5·a_5_10 + a_3_4·a_5_10
33. a_2_1·a_6_9
34. a_2_1·b_2_2³ + a_3_5·a_5_10 + a_1_1·a_7_16 + c_2_3²·a_1_1·a_3_4 + c_2_3³·a_1_1²
35. a_1_0·a_7_16
36. a_2_1·b_2_2³ + a_3_5·a_5_10 + a_1_1·a_7_19 + c_2_3·a_1_1·a_5_10 + c_2_3³·a_1_1²
37. a_1_0·a_7_19
38. a_6_9·a_3_5 + a_4_5·a_5_9 + c_2_3²·a_4_5·a_1_1
39. a_6_9·a_3_4 + c_2_3·a_4_5·a_1_1³
40. b_2_2·a_7_16 + b_2_2³·a_3_5 + a_4_5·a_5_10 + b_2_2²·a_4_5·a_1_1 + b_2_2²·c_2_3·a_3_5
       + b_2_2·c_2_3·a_4_5·a_1_1 + c_2_3·a_1_1²·a_5_10 + c_2_3·a_4_5·a_1_1³
       + c_2_3²·a_4_5·a_1_1 + b_2_2·c_2_3³·a_1_1
41. a_2_1·a_7_16
42. a_4_5·a_5_9 + a_1_1²·a_7_16 + c_2_3²·a_4_5·a_1_1 + c_2_3²·a_1_1²·a_3_4
       + c_2_3³·a_1_1³
43. a_4_5·a_5_9 + a_2_1·a_7_19 + c_2_3·a_4_5·a_1_1³ + c_2_3²·a_4_5·a_1_1
44. a_5_9² + c_2_3⁴·a_1_1²
45. a_5_9·a_5_10 + b_2_2·a_3_5·a_5_10 + c_2_3²·a_1_1·a_5_10
46. a_4_5·a_6_9 + b_2_2·c_2_3·a_4_5·a_1_1²
47. a_3_5·a_7_16 + c_2_3³·a_1_1·a_3_4
48. a_3_4·a_7_16 + c_2_3³·a_1_1·a_3_4
49. a_3_5·a_7_19 + b_2_2²·a_4_5·a_1_1² + c_2_3·a_3_5·a_5_10 + c_2_3³·a_1_1·a_3_4
50. a_3_4·a_7_19 + c_2_3·a_1_1·a_7_16 + c_2_3⁴·a_1_1²
51. a_5_10² + b_2_2·a_3_5·a_5_10 + b_2_2²·a_1_1·a_5_10 + b_2_2⁴·a_1_1²
       + a_4_5·a_1_1·a_5_10 + b_2_2²·a_4_5·a_1_1² + c_8_24·a_1_1²
       + c_2_3²·a_4_5·a_1_1²
52. a_6_9·a_5_9 + a_4_5·a_1_1²·a_5_10 + c_2_3³·a_1_1²·a_3_4
53. a_6_9·a_5_10 + a_4_5·a_7_16 + a_4_5·a_1_1²·a_5_10 + c_2_3³·a_4_5·a_1_1
54. a_6_9·a_5_10 + a_4_5·a_1_1²·a_5_10 + c_2_3·a_1_1²·a_7_16
       + b_2_2·c_2_3·a_1_1²·a_5_10 + c_2_3³·a_1_1²·a_3_4 + c_2_3⁴·a_1_1³
55. a_4_5·a_7_19 + a_4_5·a_1_1²·a_5_10 + c_2_3·a_4_5·a_5_10 + b_2_2·c_2_3·a_1_1²·a_5_10
       + c_2_3²·a_4_5·a_1_1³ + c_2_3³·a_4_5·a_1_1
56. a_6_9²
57. a_5_9·a_7_16 + c_2_3²·a_1_1·a_7_16 + c_2_3³·a_4_5·a_1_1²
58. a_5_9·a_7_19 + a_5_9·a_7_16 + b_2_2³·a_4_5·a_1_1² + b_2_2·c_2_3·a_3_5·a_5_10
       + c_2_3³·a_1_1·a_5_10 + c_2_3⁴·a_1_1·a_3_4
59. a_5_10·a_7_16 + a_5_9·a_7_16 + b_2_2²·a_3_5·a_5_10 + b_2_2³·a_4_5·a_1_1²
       + c_8_24·a_1_1·a_3_4 + b_2_2·c_2_3·a_3_5·a_5_10 + c_2_3·a_4_5·a_1_1·a_5_10
       + c_2_3³·a_1_1·a_5_10 + c_2_3³·a_4_5·a_1_1² + c_2_3⁴·a_1_1·a_3_4
       + c_2_3⁵·a_1_1²
60. b_2_2³·a_6_9 + a_5_10·a_7_19 + a_5_10·a_7_16 + a_5_9·a_7_16 + b_2_2·c_2_3·a_3_5·a_5_10
       + b_2_2²·c_2_3·a_1_1·a_5_10 + c_2_3·a_4_5·a_1_1·a_5_10 + c_2_3·c_8_24·a_1_1²
       + c_2_3⁴·a_1_1·a_3_4 + c_2_3⁵·a_1_1²
61. a_6_9·a_7_16 + b_2_2·a_4_5·a_1_1²·a_5_10 + c_2_3³·a_4_5·a_1_1³
       + c_2_3⁴·a_1_1²·a_3_4
62. a_6_9·a_7_19 + b_2_2·a_4_5·a_1_1²·a_5_10 + c_8_24·a_1_1²·a_3_4
       + c_2_3·a_4_5·a_1_1²·a_5_10 + c_2_3²·a_1_1²·a_7_16 + b_2_2·c_2_3²·a_1_1²·a_5_10
       + c_2_3⁵·a_1_1³
63. a_7_16² + c_2_3⁶·a_1_1²
64. a_7_16·a_7_19 + b_2_2⁴·a_4_5·a_1_1² + b_2_2²·c_2_3·a_3_5·a_5_10
       + c_2_3·c_8_24·a_1_1·a_3_4 + b_2_2·c_2_3²·a_3_5·a_5_10 + c_2_3²·a_4_5·a_1_1·a_5_10
       + c_2_3³·a_1_1·a_7_16 + c_2_3⁴·a_1_1·a_5_10
65. a_7_19² + b_2_2·c_2_3²·a_3_5·a_5_10 + b_2_2²·c_2_3²·a_1_1·a_5_10
       + b_2_2⁴·c_2_3²·a_1_1² + c_2_3²·a_4_5·a_1_1·a_5_10
       + b_2_2²·c_2_3²·a_4_5·a_1_1² + c_2_3²·c_8_24·a_1_1² + c_2_3⁴·a_4_5·a_1_1²
       + c_2_3⁶·a_1_1²

About the group

Ring generators

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 14.
• 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_3, a Duflot regular element of degree 2
  2. c_8_24, a Duflot regular element of degree 8
  3. b_2_2, 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

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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. a_1_1 → 0, an element of degree 1
3. a_2_1 → 0, an element of degree 2
4. b_2_2 → 0, an element of degree 2
5. c_2_3 → c_1_0², an element of degree 2
6. a_3_4 → 0, an element of degree 3
7. a_3_5 → 0, an element of degree 3
8. a_4_5 → 0, an element of degree 4
9. a_5_9 → 0, an element of degree 5
10. a_5_10 → 0, an element of degree 5
11. a_6_9 → 0, an element of degree 6
12. a_7_16 → 0, an element of degree 7
13. a_7_19 → 0, an element of degree 7
14. c_8_24 → c_1_1⁸, 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_2_1 → 0, an element of degree 2
4. b_2_2 → c_1_2², an element of degree 2
5. c_2_3 → c_1_0², an element of degree 2
6. a_3_4 → 0, an element of degree 3
7. a_3_5 → 0, an element of degree 3
8. a_4_5 → 0, an element of degree 4
9. a_5_9 → 0, an element of degree 5
10. a_5_10 → 0, an element of degree 5
11. a_6_9 → 0, an element of degree 6
12. a_7_16 → 0, an element of degree 7
13. a_7_19 → 0, an element of degree 7
14. c_8_24 → c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0²·c_1_2⁶ + c_1_0⁴·c_1_2⁴, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128