Simon King
David J. Green
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Cohomology of group number 526 of order 128
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 9 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 3.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
( − 1) · (t2 + 1)2 |
| (t − 1)3 · (t4 + 1) |
- The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_2_3, an element of degree 2
- b_2_4, an element of degree 2
- b_2_5, an element of degree 2
- b_2_6, an element of degree 2
- b_2_7, an element of degree 2
- b_3_14, an element of degree 3
- b_3_15, an element of degree 3
- b_5_39, an element of degree 5
- c_8_88, a Duflot regular element of degree 8
Ring relations
There are 36 minimal relations of maximal degree 10:
- a_1_02
- a_1_0·b_1_1
- a_1_0·b_1_2
- b_2_3·a_1_0
- b_2_4·a_1_0
- b_2_4·b_1_1 + b_2_3·b_1_2
- b_2_5·a_1_0
- b_2_7·b_1_1 + b_2_6·b_1_2 + b_2_6·b_1_1 + b_2_6·a_1_0
- b_2_5·b_1_22 + b_2_5·b_1_1·b_1_2 + b_2_52 + b_2_42 + b_2_3·b_2_4 + b_2_32
- b_2_6·b_1_22 + b_2_42
- b_2_6·b_1_1·b_1_2 + b_2_3·b_2_4
- b_2_6·b_1_12 + b_2_32
- b_1_2·b_3_14 + b_2_4·b_2_5 + b_2_42 + b_2_3·b_2_4
- a_1_0·b_3_14
- b_1_1·b_3_14 + b_2_3·b_2_5 + b_2_3·b_2_4 + b_2_32
- b_1_2·b_3_15 + b_2_5·b_2_7 + b_2_5·b_2_6 + b_2_4·b_2_6 + b_2_4·b_2_5 + b_2_3·b_2_4
- b_2_4·b_2_6 + b_2_3·b_2_7 + b_2_3·b_2_6 + a_1_0·b_3_15
- b_1_1·b_3_15 + b_2_5·b_2_6 + b_2_3·b_2_6 + b_2_3·b_2_5 + b_2_32
- b_2_6·b_2_7·b_1_2 + b_2_3·b_2_4·b_1_2 + b_2_32·b_1_2 + b_2_6·b_2_7·a_1_0
- b_2_5·b_2_6·b_1_1 + b_2_3·b_3_14 + b_2_3·b_2_6·b_1_2 + b_2_3·b_2_6·b_1_1
- b_2_5·b_3_14 + b_2_5·b_2_6·b_1_2 + b_2_5·b_2_6·b_1_1 + b_2_4·b_2_5·b_1_2
+ b_2_3·b_2_7·b_1_2 + b_2_3·b_2_6·b_1_1 + b_2_3·b_2_5·b_1_2
- b_2_5·b_2_6·b_1_2 + b_2_4·b_3_14 + b_2_3·b_2_7·b_1_2
- b_2_6·b_3_14 + b_2_6·b_2_7·b_1_2 + b_2_62·b_1_2 + b_2_5·b_2_6·b_1_1 + b_2_3·b_3_15
+ b_2_3·b_2_6·b_1_1 + b_2_3·b_2_4·b_1_2 + b_2_32·b_1_2
- b_2_6·b_3_14 + b_2_6·b_2_7·b_1_2 + b_2_62·b_1_2 + b_2_5·b_3_15 + b_2_5·b_2_7·b_1_2
+ b_2_5·b_2_6·b_1_1 + b_2_4·b_2_5·b_1_2 + b_2_3·b_2_7·b_1_2 + b_2_3·b_2_6·b_1_1 + b_2_3·b_2_5·b_1_2
- b_2_7·b_3_14 + b_2_6·b_3_14 + b_2_6·b_2_7·b_1_2 + b_2_62·b_1_2 + b_2_5·b_2_6·b_1_2
+ b_2_4·b_3_15 + b_2_3·b_2_6·b_1_2
- b_3_142 + b_2_42·b_2_5 + b_2_3·b_2_4·b_2_5 + b_2_32·b_2_7 + b_2_32·b_2_6
- b_3_14·b_3_15 + b_2_5·b_2_6·b_2_7 + b_2_5·b_2_62 + b_2_4·b_2_5·b_2_7 + b_2_42·b_2_5
+ b_2_3·b_2_72 + b_2_3·b_2_62 + b_2_3·b_2_5·b_2_7 + b_2_3·b_2_5·b_2_6 + b_2_3·b_2_4·b_2_5 + b_2_32·b_1_22 + b_2_32·b_1_1·b_1_2 + b_2_32·b_2_7 + b_2_32·b_2_6 + b_2_7·a_1_0·b_3_15 + b_2_6·a_1_0·b_3_15
- b_3_152 + b_2_6·b_2_72 + b_2_62·b_2_7 + b_2_5·b_2_72 + b_2_5·b_2_6·b_2_7
+ b_2_42·b_2_5 + b_2_3·b_2_72 + b_2_3·b_2_6·b_2_7 + b_2_3·b_2_4·b_2_5 + b_2_3·b_2_42 + b_2_32·b_1_22 + b_2_32·b_1_1·b_1_2 + b_2_32·b_2_4
- b_1_2·b_5_39 + b_2_4·b_2_5·b_2_7 + b_2_3·b_2_72 + b_2_3·b_2_6·b_2_7
+ b_2_3·b_2_5·b_1_1·b_1_2 + b_2_3·b_2_5·b_2_7 + b_2_3·b_2_5·b_2_6 + b_2_3·b_2_52 + b_2_3·b_2_4·b_1_22 + b_2_3·b_2_4·b_2_5 + b_2_32·b_1_1·b_1_2 + b_2_32·b_2_4 + b_2_33 + b_2_7·a_1_0·b_3_15
- b_2_3·b_2_72 + b_2_3·b_2_6·b_2_7 + b_2_3·b_2_42 + b_2_32·b_2_4 + a_1_0·b_5_39
+ b_2_7·a_1_0·b_3_15
- b_1_1·b_5_39 + b_2_5·b_2_6·b_2_7 + b_2_3·b_2_6·b_2_7 + b_2_3·b_2_5·b_1_1·b_1_2
+ b_2_3·b_2_5·b_2_7 + b_2_3·b_2_5·b_2_6 + b_2_3·b_2_4·b_2_5 + b_2_32·b_1_22 + b_2_32·b_1_1·b_1_2 + b_2_32·b_2_7 + b_2_32·b_2_4
- b_2_3·b_5_39 + b_2_3·b_2_7·b_3_15 + b_2_3·b_2_62·b_1_1 + b_2_3·b_2_4·b_3_14
+ b_2_32·b_3_15 + b_2_32·b_3_14 + b_2_32·b_2_7·b_1_2 + b_2_32·b_2_5·b_1_2 + b_2_32·b_2_4·b_1_2 + b_2_33·b_1_2 + b_2_6·b_2_72·a_1_0 + b_2_62·b_2_7·a_1_0
- b_2_5·b_5_39 + b_2_4·b_2_5·b_2_7·b_1_2 + b_2_3·b_2_7·b_3_15 + b_2_3·b_2_62·b_1_1
+ b_2_3·b_2_52·b_1_2 + b_2_3·b_2_4·b_3_14 + b_2_3·b_2_42·b_1_2 + b_2_32·b_2_6·b_1_2 + b_2_32·b_2_4·b_1_2 + b_2_6·b_2_72·a_1_0 + b_2_62·b_2_7·a_1_0
- b_2_4·b_5_39 + b_2_3·b_2_62·b_1_2 + b_2_3·b_2_4·b_3_15 + b_2_3·b_2_42·b_1_2
+ b_2_32·b_2_5·b_1_2 + b_2_33·b_1_2 + b_2_6·b_2_72·a_1_0 + b_2_62·b_2_7·a_1_0
- b_3_14·b_5_39 + b_2_3·b_2_5·b_2_6·b_2_7 + b_2_32·b_2_7·b_1_22 + b_2_32·b_2_62
+ b_2_32·b_2_5·b_2_7 + b_2_32·b_2_5·b_2_6 + b_2_33·b_1_22 + b_2_33·b_1_1·b_1_2 + b_2_33·b_2_7 + b_2_33·b_2_6 + b_2_33·b_2_4 + b_2_6·a_1_0·b_5_39 + b_2_6·b_2_7·a_1_0·b_3_15
- b_5_392 + b_2_62·b_2_73 + b_2_63·b_2_72 + b_2_32·b_2_63
+ b_2_32·b_2_42·b_1_22 + b_2_32·b_2_42·b_2_5 + b_2_33·b_2_4·b_1_22 + b_2_33·b_2_4·b_2_5 + b_2_34·b_1_22
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- 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:
- c_8_88, a Duflot regular element of degree 8
- b_1_24 + b_1_12·b_1_22 + b_1_14 + b_2_72 + b_2_6·b_2_7 + b_2_62 + b_2_5·b_2_6
+ b_2_4·b_2_7 + b_2_3·b_2_6 + b_2_3·b_2_5, an element of degree 4
- b_1_12·b_1_24 + b_1_14·b_1_22 + b_2_72·b_1_22 + b_2_6·b_2_72 + b_2_62·b_2_7
+ b_2_4·b_2_7·b_1_22 + b_2_4·b_2_72 + b_2_4·b_2_5·b_2_7 + b_2_3·b_2_5·b_1_1·b_1_2 + b_2_3·b_2_5·b_1_12 + b_2_3·b_2_5·b_2_6 + b_2_3·b_2_52 + b_2_3·b_2_4·b_2_5 + b_2_3·b_2_42 + b_2_32·b_1_1·b_1_2 + b_2_33, an element of degree 6
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 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.
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_2_6 → 0, an element of degree 2
- b_2_7 → 0, an element of degree 2
- b_3_14 → 0, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → 0, an element of degree 5
- c_8_88 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_2_6 → 0, an element of degree 2
- b_2_7 → 0, an element of degree 2
- b_3_14 → 0, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → 0, an element of degree 5
- c_8_88 → 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_2_6 → 0, an element of degree 2
- b_2_7 → 0, an element of degree 2
- b_3_14 → 0, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → 0, an element of degree 5
- c_8_88 → 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_3 → c_1_1·c_1_2, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → c_1_1·c_1_2, an element of degree 2
- b_2_6 → c_1_22, an element of degree 2
- b_2_7 → c_1_22, an element of degree 2
- b_3_14 → 0, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → c_1_1·c_1_24, an element of degree 5
- c_8_88 → 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_2_6 → 0, an element of degree 2
- b_2_7 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_3_14 → 0, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → 0, an element of degree 5
- c_8_88 → 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- b_2_3 → c_1_22, an element of degree 2
- b_2_4 → c_1_1·c_1_2, an element of degree 2
- b_2_5 → c_1_22, an element of degree 2
- b_2_6 → c_1_22, an element of degree 2
- b_2_7 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_3_14 → c_1_1·c_1_22, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → c_1_25 + c_1_12·c_1_23, an element of degree 5
- c_8_88 → c_1_1·c_1_27 + 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → c_1_22, an element of degree 2
- b_2_6 → 0, an element of degree 2
- b_2_7 → c_1_1·c_1_2 + c_1_12, an element of degree 2
- b_3_14 → 0, an element of degree 3
- b_3_15 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_5_39 → 0, an element of degree 5
- c_8_88 → c_1_12·c_1_26 + c_1_14·c_1_24 + 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_2_6 → c_1_22, an element of degree 2
- b_2_7 → c_1_12, an element of degree 2
- b_3_14 → 0, an element of degree 3
- b_3_15 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_5_39 → c_1_12·c_1_23 + c_1_13·c_1_22, an element of degree 5
- c_8_88 → c_1_28 + c_1_12·c_1_26 + 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- b_2_3 → c_1_1·c_1_2, an element of degree 2
- b_2_4 → c_1_1·c_1_2, an element of degree 2
- b_2_5 → c_1_1·c_1_2, an element of degree 2
- b_2_6 → c_1_22, an element of degree 2
- b_2_7 → 0, an element of degree 2
- b_3_14 → c_1_1·c_1_22, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
- c_8_88 → c_1_28 + c_1_1·c_1_27 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
+ 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
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → c_1_2 + c_1_1, an element of degree 1
- b_2_3 → c_1_22, an element of degree 2
- b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_2_5 → c_1_22 + c_1_1·c_1_2 + c_1_12, an element of degree 2
- b_2_6 → c_1_22, an element of degree 2
- b_2_7 → c_1_1·c_1_2, an element of degree 2
- b_3_14 → c_1_23 + c_1_12·c_1_2, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_5_39 → c_1_1·c_1_24, an element of degree 5
- c_8_88 → c_1_12·c_1_26 + 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
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