Cohomology of group number 1958 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 3 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 coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t6  +  t5  +  t2  +  t  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-5,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

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

  1. a_1_2, a nilpotent element of degree 1
  2. b_1_0, an element of degree 1
  3. b_1_1, an element of degree 1
  4. b_1_3, an element of degree 1
  5. c_2_8, a Duflot regular element of degree 2
  6. a_5_27, a nilpotent element of degree 5
  7. b_5_26, an element of degree 5
  8. c_8_56, a Duflot regular element of degree 8

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Ring relations

There are 10 minimal relations of maximal degree 10:

  1. a_1_2·b_1_0
  2. b_1_12 + b_1_0·b_1_3 + b_1_0·b_1_1 + a_1_2·b_1_1 + a_1_22
  3. b_1_0·b_1_32 + a_1_23
  4. a_1_23·b_1_32
  5. b_1_04·b_1_1·b_1_3 + b_1_0·a_5_27
  6. a_1_2·b_5_26 + c_2_8·a_1_2·b_1_33 + c_2_8·a_1_22·b_1_32 + c_2_8·a_1_23·b_1_1
  7. b_1_32·b_5_26 + a_1_22·a_5_27 + c_2_8·b_1_35 + c_2_8·a_1_2·b_1_34
       + c_2_8·a_1_22·b_1_1·b_1_32
  8. b_1_03·b_1_1·b_1_3·b_5_26 + a_5_27·b_5_26 + c_2_8·b_1_33·a_5_27
       + c_2_8·a_1_2·b_1_32·a_5_27 + c_2_8·a_1_22·b_1_1·a_5_27
  9. b_5_262 + b_1_05·b_5_26 + b_1_09·b_1_3 + b_1_05·a_5_27 + c_8_56·b_1_02
       + c_2_8·b_1_03·b_5_26 + c_2_8·b_1_07·b_1_3 + c_2_82·b_1_36
       + c_2_82·a_1_22·b_1_34 + c_2_83·b_1_03·b_1_3
  10. a_5_272 + a_1_2·b_1_34·a_5_27 + a_1_22·b_1_33·a_5_27 + c_8_56·a_1_22
       + c_2_8·a_1_22·b_1_36 + c_2_8·a_1_22·b_1_1·b_1_35 + c_2_83·a_1_23·b_1_3
       + c_2_83·a_1_23·b_1_1

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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:
    1. c_2_8, a Duflot regular element of degree 2
    2. c_8_56, a Duflot regular element of degree 8
    3. b_1_32 + b_1_02, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 9].
  • 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_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_2_8 → c_1_12, an element of degree 2
  6. a_5_27 → 0, an element of degree 5
  7. b_5_26 → 0, an element of degree 5
  8. c_8_56 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → c_1_2, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_2_8 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  6. a_5_27 → 0, an element of degree 5
  7. b_5_26 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  8. c_8_56 → c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_24
       + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. c_2_8 → c_1_12, an element of degree 2
  6. a_5_27 → 0, an element of degree 5
  7. b_5_26 → c_1_12·c_1_23, an element of degree 5
  8. c_8_56 → c_1_12·c_1_26 + c_1_04·c_1_24 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → c_1_2, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_2_8 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  6. a_5_27 → 0, an element of degree 5
  7. b_5_26 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  8. c_8_56 → c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_24
       + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + 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