Cohomology of group number 2312 of order 128

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General information on the group

  • The group has 5 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    t6  +  t5  +  t2  +  t  +  1
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-∞,-6,-4. 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. 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. b_1_3, an element of degree 1
  5. c_1_4, a Duflot regular element of degree 1
  6. b_5_76, an element of degree 5
  7. b_5_77, an element of degree 5
  8. c_8_219, a Duflot regular element of degree 8

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Ring relations

There are 9 minimal relations of maximal degree 10:

  1. b_1_0·b_1_1
  2. b_1_1·b_1_32 + b_1_0·b_1_32 + b_1_0·b_1_22
  3. b_1_0·b_1_22·b_1_32 + b_1_0·b_1_24
  4. b_1_1·b_5_76
  5. b_1_0·b_5_77
  6. b_1_32·b_5_77 + b_1_32·b_5_76 + b_1_22·b_5_76
  7. b_5_76·b_5_77 + b_1_24·b_1_36 + b_1_26·b_1_34
  8. b_5_762 + b_1_24·b_1_36 + b_1_0·b_1_24·b_5_76 + b_1_0·b_1_29 + b_1_04·b_1_26
       + c_8_219·b_1_02
  9. b_5_772 + b_1_24·b_1_36 + b_1_28·b_1_32 + b_1_1·b_1_24·b_5_77
       + b_1_12·b_1_23·b_5_77 + b_1_14·b_1_2·b_5_77 + c_8_219·b_1_12

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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_1_4, a Duflot regular element of degree 1
    2. c_8_219, a Duflot regular element of degree 8
    3. b_1_32 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_2 + b_1_12 + b_1_0·b_1_2 + b_1_02, an element of degree 2
    4. b_1_2·b_1_32 + b_1_22·b_1_3 + b_1_1·b_1_22 + b_1_12·b_1_2 + b_1_0·b_1_22
         + b_1_02·b_1_3, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 10].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

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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. b_1_3 → 0, an element of degree 1
  5. c_1_4 → c_1_0, an element of degree 1
  6. b_5_76 → 0, an element of degree 5
  7. b_5_77 → 0, an element of degree 5
  8. c_8_219 → c_1_18, an element of degree 8

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

  1. b_1_0 → c_1_2, 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. b_1_3 → c_1_3, an element of degree 1
  5. c_1_4 → c_1_0, an element of degree 1
  6. b_5_76 → c_1_35 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  7. b_5_77 → 0, an element of degree 5
  8. c_8_219 → c_1_22·c_1_36 + c_1_12·c_1_22·c_1_34 + c_1_14·c_1_34 + c_1_14·c_1_24
       + c_1_18, an element of degree 8

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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_1_4 → c_1_0, an element of degree 1
  6. b_5_76 → 0, an element of degree 5
  7. b_5_77 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  8. c_8_219 → c_1_12·c_1_22·c_1_34 + c_1_12·c_1_23·c_1_33 + c_1_12·c_1_25·c_1_3
       + c_1_14·c_1_34 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_23·c_1_3 + c_1_14·c_1_24
       + c_1_18, an element of degree 8

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

  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_2, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. c_1_4 → c_1_0, an element of degree 1
  6. b_5_76 → c_1_22·c_1_33, an element of degree 5
  7. b_5_77 → c_1_22·c_1_33 + c_1_24·c_1_3, an element of degree 5
  8. c_8_219 → c_1_38 + c_1_23·c_1_35 + c_1_24·c_1_34 + c_1_27·c_1_3
       + c_1_12·c_1_22·c_1_34 + c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34
       + c_1_14·c_1_22·c_1_32 + c_1_14·c_1_24 + c_1_18, 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