Mod-2-Cohomology of group number 1088539 of order 768

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General information on the group

  • The group order factors as 28 · 3.
  • It is non-abelian.
  • It has 2-Rank 5.
  • The centre of a Sylow 2-subgroup has rank 2.
  • Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4, 4, 5, 5 and 5, respectively.

Structure of the cohomology ring

The computation was based on 1 stability condition for H*(SmallGroup(256,26531); GF(2)).

General information

  • The cohomology ring is of dimension 5 and depth 4.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1)·(1  +  2·t2  +  t3  +  t5)
    (1  +  t) · ( − 1  +  t)5 · (1  +  t2) · (1  +  t  +  t2)
  • The a-invariants are -∞,-∞,-∞,-∞,-7,-5. They were obtained using the filter regular HSOP of the Benson test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 4:

  1. c_1_2, a Duflot element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_0, an element of degree 1
  4. c_2_5, a Duflot element of degree 2
  5. b_2_4, an element of degree 2
  6. c_3_15, a Duflot element of degree 3
  7. b_3_1, an element of degree 3
  8. b_3_0, an element of degree 3
  9. c_4_24, a Duflot element of degree 4

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Ring relations

There are 10 minimal relations of maximal degree 6:

  1. c_2_5·b_1_1 + b_1_1·c_1_22
  2. b_1_0·c_3_15 + c_2_5·b_1_0·c_1_2
  3. b_1_1·c_3_15 + b_1_1·c_1_23
  4. b_1_1·b_3_0 + b_1_0·b_3_1 + b_2_4·b_1_1·c_1_2
  5. b_2_4·c_3_15 + b_2_4·c_2_5·c_1_2
  6. c_2_5·b_3_1 + c_1_22·b_3_1
  7. b_3_0·c_3_15 + c_2_5·c_1_2·b_3_0
  8. b_3_1·c_3_15 + c_1_23·b_3_1
  9. b_3_02 + b_2_4·b_1_0·b_3_0 + c_4_24·b_1_02 + b_2_42·b_1_0·c_1_2 + b_2_42·c_2_5
  10. b_3_0·b_3_1 + b_2_4·b_1_0·b_3_1 + c_4_24·b_1_0·b_1_1 + b_2_4·c_1_2·b_3_1

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Data used for the Benson test

  • We proved completion in degree 19 using the Benson criterion.
  • However, the last relation was already found in degree 6 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_1_2, an element of degree 1
    2. c_4_24, an element of degree 4
    3. b_1_1·b_3_1 + b_1_14 + b_1_0·b_3_0 + b_1_02·b_1_12 + b_1_04 + b_2_4·b_1_0·b_1_1
         + b_2_4·b_1_02 + b_2_42 + b_2_4·b_1_0·c_1_2 + c_2_52 + c_1_24, an element of degree 4
    4. b_3_12 + b_1_13·b_3_1 + b_1_0·b_1_12·b_3_1 + b_1_02·b_1_14 + b_1_03·b_3_1
         + b_1_03·b_3_0 + b_1_04·b_1_12 + b_2_4·b_1_1·b_3_1 + b_2_4·b_1_0·b_1_13
         + b_2_4·b_1_04 + b_2_42·b_1_12 + b_2_42·b_1_0·b_1_1 + b_2_42·b_1_02
         + c_4_24·b_1_12 + c_2_5·b_1_0·b_3_0 + b_2_4·b_1_03·c_1_2 + b_2_4·c_2_5·b_1_02
         + b_2_42·c_2_5 + c_3_152 + b_1_0·c_1_22·b_3_0 + c_2_52·b_1_02
         + b_2_4·b_1_02·c_1_22 + b_2_4·c_2_5·b_1_0·c_1_2 + b_2_42·c_1_22
         + b_2_4·b_1_0·c_1_23 + b_1_02·c_1_24 + c_2_52·c_1_22, an element of degree 6
    5. b_1_1·b_3_12 + b_1_0·b_1_13·b_3_1 + b_1_02·b_1_12·b_3_1 + b_1_03·b_1_1·b_3_1
         + b_1_04·b_3_1 + b_2_4·b_1_12·b_3_1 + b_2_4·b_1_02·b_1_13 + b_2_4·b_1_04·b_1_1
         + b_2_42·b_1_0·b_1_12 + b_2_42·b_1_02·b_1_1 + c_4_24·b_1_13
         + c_2_5·b_1_02·b_3_0 + b_2_4·c_2_5·b_1_03 + b_2_42·c_2_5·b_1_0
         + b_1_02·c_1_22·b_3_0 + b_2_4·b_1_03·c_1_22 + b_2_4·c_2_5·b_1_02·c_1_2
         + b_2_42·b_1_0·c_1_22 + b_2_4·b_1_02·c_1_23, an element of degree 7
  • A Duflot regular sequence is given by c_1_2, c_4_24.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, -1, 8, 17].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(256,26531); GF(2))

  1. c_1_2 → b_1_0 + c_1_3
  2. b_1_1 → b_1_1
  3. b_1_0 → b_1_2
  4. c_2_5 → b_2_9 + c_1_32
  5. b_2_4 → b_2_10 + b_2_8
  6. c_3_15 → b_2_9·c_1_3 + b_1_0·c_1_32 + c_1_33
  7. b_3_1 → b_3_22
  8. b_3_0 → b_3_23 + b_2_10·c_1_3 + b_2_8·c_1_3
  9. c_4_24 → c_4_45

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. c_1_2 → c_1_0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. c_2_5 → c_1_02, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. c_3_15 → c_1_03, an element of degree 3
  7. b_3_1 → 0, an element of degree 3
  8. b_3_0 → 0, an element of degree 3
  9. c_4_24 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. c_1_2 → c_1_3 + c_1_1 + c_1_0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. c_2_5 → c_1_12 + c_1_02, an element of degree 2
  5. b_2_4 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. c_3_15 → c_1_12·c_1_3 + c_1_13 + c_1_0·c_1_12 + c_1_02·c_1_3 + c_1_02·c_1_1 + c_1_03, an element of degree 3
  7. b_3_1 → 0, an element of degree 3
  8. b_3_0 → c_1_1·c_1_2·c_1_3 + c_1_1·c_1_22 + c_1_0·c_1_2·c_1_3 + c_1_0·c_1_22, an element of degree 3
  9. c_4_24 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
       + c_1_12·c_1_22 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. c_1_2 → c_1_3 + c_1_1 + c_1_0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. c_2_5 → c_1_2·c_1_3 + c_1_22 + c_1_12 + c_1_02, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. c_3_15 → c_1_1·c_1_2·c_1_3 + c_1_1·c_1_22 + c_1_12·c_1_3 + c_1_13 + c_1_0·c_1_2·c_1_3
       + c_1_0·c_1_22 + c_1_0·c_1_12 + c_1_02·c_1_3 + c_1_02·c_1_1 + c_1_03, an element of degree 3
  7. b_3_1 → 0, an element of degree 3
  8. b_3_0 → 0, an element of degree 3
  9. c_4_24 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
       + c_1_12·c_1_22 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5 in a Sylow subgroup

  1. c_1_2 → c_1_1 + c_1_0, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_0 → c_1_4, an element of degree 1
  4. c_2_5 → c_1_12 + c_1_02, an element of degree 2
  5. b_2_4 → c_1_2·c_1_4 + c_1_22 + c_1_1·c_1_4 + c_1_1·c_1_3, an element of degree 2
  6. c_3_15 → c_1_13 + c_1_0·c_1_12 + c_1_02·c_1_1 + c_1_03, an element of degree 3
  7. b_3_1 → c_1_2·c_1_3·c_1_4 + c_1_22·c_1_3 + c_1_1·c_1_3·c_1_4 + c_1_12·c_1_3, an element of degree 3
  8. b_3_0 → c_1_2·c_1_42 + c_1_22·c_1_4 + c_1_1·c_1_42 + c_1_1·c_1_2·c_1_4 + c_1_1·c_1_22
       + c_1_12·c_1_3 + c_1_0·c_1_2·c_1_4 + c_1_0·c_1_22 + c_1_0·c_1_1·c_1_4
       + c_1_0·c_1_1·c_1_3, an element of degree 3
  9. c_4_24 → c_1_1·c_1_2·c_1_3·c_1_4 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_3·c_1_4
       + c_1_12·c_1_2·c_1_4 + c_1_12·c_1_22 + c_1_13·c_1_4 + c_1_13·c_1_3 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5 in a Sylow subgroup

  1. c_1_2 → c_1_1 + c_1_0, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. c_2_5 → c_1_12 + c_1_02, an element of degree 2
  5. b_2_4 → c_1_42 + c_1_3·c_1_4 + c_1_2·c_1_4 + c_1_22 + c_1_1·c_1_3, an element of degree 2
  6. c_3_15 → c_1_13 + c_1_0·c_1_12 + c_1_02·c_1_1 + c_1_03, an element of degree 3
  7. b_3_1 → c_1_2·c_1_42 + c_1_2·c_1_3·c_1_4 + c_1_22·c_1_4 + c_1_22·c_1_3 + c_1_12·c_1_3, an element of degree 3
  8. b_3_0 → c_1_1·c_1_42 + c_1_1·c_1_3·c_1_4 + c_1_1·c_1_2·c_1_4 + c_1_1·c_1_22 + c_1_12·c_1_3
       + c_1_0·c_1_42 + c_1_0·c_1_3·c_1_4 + c_1_0·c_1_2·c_1_4 + c_1_0·c_1_22
       + c_1_0·c_1_1·c_1_3, an element of degree 3
  9. c_4_24 → c_1_1·c_1_2·c_1_42 + c_1_1·c_1_2·c_1_3·c_1_4 + c_1_1·c_1_22·c_1_4
       + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_2·c_1_4
       + c_1_12·c_1_22 + c_1_13·c_1_3 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5 in a Sylow subgroup

  1. c_1_2 → c_1_1 + c_1_0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_0 → c_1_3, an element of degree 1
  4. c_2_5 → c_1_42 + c_1_3·c_1_4 + c_1_12 + c_1_02, an element of degree 2
  5. b_2_4 → c_1_2·c_1_4 + c_1_2·c_1_3 + c_1_22 + c_1_1·c_1_3, an element of degree 2
  6. c_3_15 → c_1_1·c_1_42 + c_1_1·c_1_3·c_1_4 + c_1_13 + c_1_0·c_1_42 + c_1_0·c_1_3·c_1_4
       + c_1_0·c_1_12 + c_1_02·c_1_1 + c_1_03, an element of degree 3
  7. b_3_1 → 0, an element of degree 3
  8. b_3_0 → c_1_2·c_1_42 + c_1_2·c_1_32 + c_1_22·c_1_4 + c_1_22·c_1_3 + c_1_1·c_1_32
       + c_1_1·c_1_2·c_1_4 + c_1_1·c_1_2·c_1_3 + c_1_1·c_1_22 + c_1_0·c_1_2·c_1_4
       + c_1_0·c_1_2·c_1_3 + c_1_0·c_1_22 + c_1_0·c_1_1·c_1_3, an element of degree 3
  9. c_4_24 → c_1_1·c_1_2·c_1_42 + c_1_1·c_1_2·c_1_3·c_1_4 + c_1_1·c_1_22·c_1_4
       + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_2·c_1_4 + c_1_12·c_1_2·c_1_3
       + c_1_12·c_1_22 + c_1_13·c_1_3 + c_1_14, an element of degree 4

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010