Cohomology of group number 1859 of order 128

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General information on the group

  • The group has 4 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
    (t2  +  t  +  1) · (t6  −  t5  −  1)
    (t  −  1)3 · (t2  +  1)2 · (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 9 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. a_1_3, a nilpotent element of degree 1
  4. b_1_1, an element of degree 1
  5. a_3_10, a nilpotent element of degree 3
  6. c_4_12, a Duflot regular element of degree 4
  7. a_5_15, a nilpotent element of degree 5
  8. b_5_16, an element of degree 5
  9. c_8_28, a Duflot regular element of degree 8

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Ring relations

There are 15 minimal relations of maximal degree 10:

  1. a_1_0·a_1_2
  2. a_1_0·b_1_1 + a_1_32
  3. a_1_32·b_1_1 + a_1_0·a_1_32
  4. a_1_3·b_1_12 + a_1_2·a_1_3·b_1_1 + a_1_33 + a_1_23 + a_1_03
  5. a_1_0·a_3_10
  6. a_1_23·b_1_12 + a_1_24·b_1_1 + a_1_25
  7. a_3_102 + a_1_2·b_1_12·a_3_10 + a_1_22·b_1_14 + a_1_22·b_1_1·a_3_10
       + c_4_12·a_1_22
  8. a_1_2·a_5_15 + a_1_2·b_1_12·a_3_10 + a_1_2·a_1_3·b_1_1·a_3_10 + a_1_22·b_1_1·a_3_10
       + a_1_25·b_1_1 + c_4_12·a_1_2·a_1_3
  9. a_1_0·b_5_16 + a_1_3·a_5_15 + a_1_0·a_5_15 + a_1_22·a_1_3·a_3_10 + a_1_23·a_3_10
       + a_1_25·b_1_1 + c_4_12·a_1_0·a_1_3 + c_4_12·a_1_02
  10. b_1_1·a_5_15 + b_1_13·a_3_10 + a_1_3·b_5_16 + a_1_3·a_5_15 + a_1_2·b_1_12·a_3_10
       + a_1_22·b_1_14 + a_1_22·b_1_1·a_3_10 + a_1_22·a_1_3·a_3_10 + a_1_23·a_3_10
       + a_1_25·b_1_1 + c_4_12·a_1_32 + c_4_12·a_1_2·a_1_3 + c_4_12·a_1_0·a_1_3
  11. a_1_3·b_1_1·b_5_16 + a_1_32·b_5_16 + a_1_22·a_1_3·b_1_1·a_3_10 + c_4_12·a_1_23
       + c_4_12·a_1_03
  12. a_3_10·a_5_15 + a_1_2·b_1_14·a_3_10 + a_1_22·b_1_16 + a_1_22·b_1_13·a_3_10
       + a_1_24·b_1_1·a_3_10 + c_4_12·a_1_3·a_3_10 + c_4_12·a_1_22·b_1_12
       + c_4_12·a_1_22·a_1_3·b_1_1 + c_4_12·a_1_23·b_1_1
  13. a_5_152 + a_1_2·b_1_16·a_3_10 + a_1_22·b_1_18 + a_1_22·b_1_15·a_3_10
       + c_8_28·a_1_02 + c_4_12·a_1_22·b_1_14 + c_4_12·a_1_25·b_1_1 + c_4_122·a_1_32
  14. a_5_15·b_5_16 + b_1_12·a_3_10·b_5_16 + a_5_152 + a_1_2·b_1_1·a_3_10·b_5_16
       + a_1_2·b_1_16·a_3_10 + a_1_22·b_1_13·b_5_16 + a_1_22·b_1_18
       + a_1_22·a_3_10·b_5_16 + a_1_22·b_1_15·a_3_10 + c_8_28·a_1_0·a_1_3
       + c_4_12·a_1_3·a_5_15 + c_4_12·a_1_22·b_1_14 + c_4_12·a_1_0·a_5_15
       + c_4_12·a_1_2·a_1_3·b_1_1·a_3_10 + c_4_12·a_1_22·a_1_3·a_3_10
       + c_4_12·a_1_25·b_1_1 + c_4_122·a_1_3·b_1_1 + c_4_122·a_1_2·a_1_3
  15. b_5_162 + b_1_17·a_3_10 + a_5_152 + a_1_2·b_1_16·a_3_10 + a_1_22·b_1_18
       + a_1_22·b_1_15·a_3_10 + c_4_12·b_1_16 + c_8_28·a_1_32 + c_4_12·a_1_22·b_1_14
       + c_4_12·a_1_25·b_1_1 + c_4_122·b_1_12 + c_4_122·a_1_32 + c_4_122·a_1_22
       + c_4_122·a_1_02

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 11.
  • However, the last relation was already found in degree 10 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_12, a Duflot regular element of degree 4
    2. c_8_28, a Duflot regular element of degree 8
    3. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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_2 → 0, an element of degree 1
  3. a_1_3 → 0, an element of degree 1
  4. b_1_1 → 0, an element of degree 1
  5. a_3_10 → 0, an element of degree 3
  6. c_4_12 → c_1_04, an element of degree 4
  7. a_5_15 → 0, an element of degree 5
  8. b_5_16 → 0, an element of degree 5
  9. c_8_28 → c_1_18 + 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. a_1_2 → 0, an element of degree 1
  3. a_1_3 → 0, an element of degree 1
  4. b_1_1 → c_1_2, an element of degree 1
  5. a_3_10 → 0, an element of degree 3
  6. c_4_12 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  7. a_5_15 → 0, an element of degree 5
  8. b_5_16 → c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
  9. c_8_28 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8

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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: 25.08.2009