David J. Green

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Cohomology of group number 880 of order 128

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Ring generators

Ring relations

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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 3.
• Its center has rank 1.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• 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.

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 8:

1. a_1_0, a nilpotent element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an 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. a_5_13, a nilpotent element of degree 5
8. b_5_18, an element of degree 5
9. b_6_24, an element of degree 6
10. c_8_37, a Duflot regular element of degree 8

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Ring relations

There are 27 minimal relations of maximal degree 12:

1. a_1_0²
2. a_1_0·b_1_1
3. b_2_4·a_1_0
4. b_1_1·b_1_2² + a_1_0·b_1_2² + b_2_5·a_1_0
5. b_2_5·b_1_1² + b_2_4²
6. b_2_4·b_1_2² + a_1_0·b_3_9
7. b_1_1·b_3_9 + b_2_4·b_1_2² + b_2_4·b_2_5
8. b_2_5·a_1_0·b_1_2² + b_2_5²·a_1_0
9. b_2_5²·b_1_1 + b_2_4·b_3_9
10. b_3_9² + b_2_5·b_1_2⁴ + b_2_5³ + b_2_5·a_1_0·b_3_9
11. b_2_5·a_1_0·b_3_9 + a_1_0·a_5_13
12. b_1_1·a_5_13
13. a_1_0·b_5_18 + b_2_5·a_1_0·b_3_9
14. b_2_4·a_5_13
15. b_1_2²·b_5_18 + b_2_5·b_1_2²·b_3_9 + b_1_2²·a_5_13 + b_2_5·a_5_13
16. b_6_24·a_1_0
17. b_6_24·b_1_1 + b_2_4·b_5_18 + b_2_4²·b_2_5·b_1_1
18. b_3_9·b_5_18 + b_2_5·b_6_24 + b_2_4²·b_2_5² + b_3_9·a_5_13 + b_2_5·a_1_0·a_5_13
19. b_6_24·b_1_2² + b_2_5·b_1_2⁶ + b_2_5³·b_1_2² + b_3_9·a_5_13 + a_1_0·b_1_2⁴·b_3_9
       + b_2_5·a_1_0·a_5_13
20. b_2_5·b_1_1·b_5_18 + b_2_4·b_6_24 + b_2_4³·b_2_5
21. b_6_24·b_3_9 + b_2_5·b_1_2⁴·b_3_9 + b_2_5²·b_5_18 + b_2_4²·b_2_5·b_3_9
       + b_2_5·b_1_2²·a_5_13 + b_2_5²·a_5_13
22. a_5_13² + b_2_5²·a_1_0·a_5_13
23. a_5_13·b_5_18 + b_2_5·b_3_9·a_5_13
24. b_5_18² + b_1_1⁵·b_5_18 + b_2_5³·b_1_2⁴ + b_2_5⁵ + b_2_4·b_1_1³·b_5_18
       + b_2_4·b_2_5·b_6_24 + b_2_4·b_2_5⁴ + b_2_4²·b_6_24 + b_2_4²·b_2_5³
       + b_2_4⁴·b_1_1² + b_2_4⁴·b_2_5 + c_8_37·b_1_1²
25. b_6_24·a_5_13 + b_2_5·b_1_2⁴·a_5_13 + b_2_5³·a_5_13
26. b_6_24·b_5_18 + b_2_5²·b_1_2⁴·b_3_9 + b_2_5⁴·b_3_9 + b_2_4·b_1_1⁴·b_5_18
       + b_2_4·b_2_5²·b_5_18 + b_2_4·b_2_5³·b_3_9 + b_2_4²·b_1_1²·b_5_18
       + b_2_4²·b_2_5²·b_3_9 + b_2_4³·b_2_5·b_3_9 + b_2_4⁵·b_1_1 + b_2_5·b_1_2⁴·a_5_13
       + b_2_5³·a_5_13 + b_2_4·c_8_37·b_1_1
27. b_6_24² + b_2_5²·b_1_2⁸ + b_2_5⁶ + b_2_4·b_2_5²·b_6_24 + b_2_4·b_2_5⁵
       + b_2_4²·b_1_1³·b_5_18 + b_2_4²·b_2_5·b_6_24 + b_2_4²·b_2_5⁴
       + b_2_4³·b_1_1·b_5_18 + b_2_4⁶ + b_2_5³·a_1_0·a_5_13 + b_2_4²·c_8_37

About the group

Ring generators

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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_8_37, a Duflot regular element of degree 8
  2. b_1_2⁴ + b_1_1⁴ + b_2_5·b_1_2² + b_2_5² + b_2_4², an element of degree 4
  3. b_2_5·b_1_2⁴ + b_2_5²·b_1_2² + b_2_4²·b_1_1² + b_2_4²·b_2_5, an element of degree 6
• The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 15].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
• We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 2.

About the group

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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. b_1_1 → 0, an element of degree 1
3. b_1_2 → 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. a_5_13 → 0, an element of degree 5
8. b_5_18 → 0, an element of degree 5
9. b_6_24 → 0, an element of degree 6
10. c_8_37 → 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. b_1_1 → c_1_1, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_4 → c_1_1·c_1_2, an element of degree 2
5. b_2_5 → c_1_2², an element of degree 2
6. b_3_9 → c_1_2³, an element of degree 3
7. a_5_13 → 0, an element of degree 5
8. b_5_18 → c_1_2⁵ + c_1_1²·c_1_2³ + c_1_1³·c_1_2² + c_1_0²·c_1_1³ + c_1_0⁴·c_1_1, an element of degree 5
9. b_6_24 → c_1_2⁶ + c_1_1³·c_1_2³ + c_1_0²·c_1_1³·c_1_2 + c_1_0⁴·c_1_1·c_1_2, an element of degree 6
10. c_8_37 → 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_1³·c_1_2³ + c_1_0²·c_1_1⁵·c_1_2 + c_1_0²·c_1_1⁶
       + c_1_0⁴·c_1_2⁴ + c_1_0⁴·c_1_1·c_1_2³ + c_1_0⁴·c_1_1³·c_1_2 + 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. b_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_2 + c_1_1, an element of degree 1
4. b_2_4 → 0, an element of degree 2
5. b_2_5 → c_1_1², an element of degree 2
6. b_3_9 → c_1_1·c_1_2², an element of degree 3
7. a_5_13 → 0, an element of degree 5
8. b_5_18 → c_1_1³·c_1_2², an element of degree 5
9. b_6_24 → c_1_1²·c_1_2⁴, an element of degree 6
10. c_8_37 → c_1_2⁸ + c_1_1²·c_1_2⁶ + 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