Cohomology of group number 26531 of order 256

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General information on the group

  • The group is also known as Syl2Sym10, the Sylow 2-subgroup of Symmetric Group Sym10.
  • The group has 4 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 5.
  • Its centre has rank 2.
  • It 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

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)
    (1  +  t) · ( − 1  +  t)5
  • The a-invariants are -∞,-∞,-∞,-∞,-5,-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 10 minimal generators of maximal degree 4:

  1. b_1_0, an element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. c_1_3, a Duflot element of degree 1
  5. b_2_8, an element of degree 2
  6. b_2_9, an element of degree 2
  7. b_2_10, an element of degree 2
  8. b_3_22, an element of degree 3
  9. b_3_23, an element of degree 3
  10. c_4_45, a Duflot element of degree 4

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Ring relations

There are 14 minimal relations of maximal degree 6:

  1. b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. b_2_8·b_1_2
  4. b_2_9·b_1_1
  5. b_2_10·b_1_0
  6. b_2_8·b_2_9
  7. b_1_0·b_3_22
  8. b_1_0·b_3_23
  9. b_1_2·b_3_22 + b_1_1·b_3_23
  10. b_2_9·b_3_22
  11. b_2_8·b_3_23
  12. b_3_232 + b_2_10·b_1_2·b_3_23 + b_2_9·b_2_102 + c_4_45·b_1_22
  13. b_3_22·b_3_23 + b_2_10·b_1_1·b_3_23 + c_4_45·b_1_1·b_1_2
  14. b_3_222 + b_2_10·b_1_1·b_3_22 + b_2_8·b_2_102 + c_4_45·b_1_12

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Data used for the Benson test

  • We proved completion in degree 7 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_4_45, an element of degree 4
    2. c_1_3, an element of degree 1
    3. b_1_2·b_3_23 + b_1_24 + b_1_1·b_3_22 + b_1_12·b_1_22 + b_1_14 + b_1_04
         + b_2_10·b_1_22 + b_2_10·b_1_1·b_1_2 + b_2_102 + b_2_92 + b_2_82, an element of degree 4
    4. b_1_23·b_3_23 + b_1_1·b_1_22·b_3_23 + b_1_12·b_1_24 + b_1_13·b_3_23
         + b_1_13·b_3_22 + b_1_14·b_1_22 + b_2_10·b_1_24 + b_2_10·b_1_13·b_1_2
         + b_2_102·b_1_22 + b_2_102·b_1_1·b_1_2 + b_2_102·b_1_12 + b_2_9·b_1_2·b_3_23
         + b_2_9·b_2_10·b_1_22 + b_2_9·b_2_102 + b_2_92·b_1_22 + b_2_92·b_1_02
         + b_2_8·b_1_1·b_3_22 + b_2_8·b_2_102 + b_2_82·b_1_12 + b_2_82·b_1_02, an element of degree 6
    5. b_1_2 + b_1_1, an element of degree 1
  • A Duflot regular sequence is given by c_4_45, c_1_3.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, -1, 10, 11].
  • We found that there exists some filter regular HSOP over a finite extension field, formed by the first 2 terms of the above HSOP, together with 3 elements of degree 2.

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. b_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. c_1_3 → c_1_0, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_2_10 → 0, an element of degree 2
  8. b_3_22 → 0, an element of degree 3
  9. b_3_23 → 0, an element of degree 3
  10. c_4_45 → c_1_14, an element of degree 4

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

  1. b_1_0 → c_1_3, 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. c_1_3 → c_1_1 + c_1_0, an element of degree 1
  5. b_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_2_10 → 0, an element of degree 2
  8. b_3_22 → 0, an element of degree 3
  9. b_3_23 → 0, an element of degree 3
  10. c_4_45 → 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

  1. b_1_0 → c_1_3, 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. c_1_3 → c_1_1 + c_1_0, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_2_9 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  7. b_2_10 → 0, an element of degree 2
  8. b_3_22 → 0, an element of degree 3
  9. b_3_23 → 0, an element of degree 3
  10. c_4_45 → 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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_2 → c_1_4, an element of degree 1
  4. c_1_3 → c_1_1 + c_1_0, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_2_10 → 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
  8. b_3_22 → 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
  9. b_3_23 → c_1_2·c_1_42 + c_1_22·c_1_4 + c_1_1·c_1_42 + c_1_12·c_1_4, an element of degree 3
  10. c_4_45 → 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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. c_1_3 → c_1_1 + c_1_0, an element of degree 1
  5. b_2_8 → c_1_42 + c_1_3·c_1_4, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_2_10 → c_1_2·c_1_4 + c_1_22 + c_1_1·c_1_3, an element of degree 2
  8. b_3_22 → 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
  9. b_3_23 → 0, an element of degree 3
  10. c_4_45 → 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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. c_1_3 → c_1_1 + c_1_0, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_2_9 → c_1_42 + c_1_3·c_1_4, an element of degree 2
  7. b_2_10 → 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
  8. b_3_22 → 0, an element of degree 3
  9. b_3_23 → 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_12·c_1_3, an element of degree 3
  10. c_4_45 → 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 A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 11.06.2010