# Second cohomology group for trivial group action of E8 on V4

From Groupprops

This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group elementary abelian group:E8 on Klein four-group. The elements of this classify the group extensions with Klein four-group in the center and elementary abelian group:E8 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.

The value of this cohomology group is elementary abelian group:E4096.

Get more specific information about elementary abelian group:E8 |Get more specific information about Klein four-group|View other constructions whose value is elementary abelian group:E4096

## Description of the group

This group is defined as the second cohomology group for trivial group action of elementary abelian group:E8 on Klein four-group, i.e., the group:

where and .

The group is an elementary abelian group of order .

## Elements

Cohomology class type | Number of cohomology classes | Corresponding group extension | GAP ID (second part, order is 32) | Base characteristic in whole group? |
---|---|---|---|---|

trivial | 1 | elementary abelian group:E32 | 51 | No |

symmetric nontrivial | direct product of Z4 and Z4 and Z2 | 21 | Yes | |

symmetric nontrivial | direct product of E8 and Z4 | 45 | No | |

non-symmetric | direct product of SmallGroup(16,3) and Z2 | 22 | Yes | |

non-symmetric | direct product of SmallGroup(16,4) and Z2 | 23 | Yes | |

non-symmetric | SmallGroup(32,24) | 24 | Yes | |

non-symmetric | direct product of D8 and Z4 | 25 | Yes | |

non-symmetric | direct product of Q8 and Z4 | 26 | Yes | |

non-symmetric | SmallGroup(32,27) | 27 | Yes | |

non-symmetric | SmallGroup(32,28) | 28 | Yes | |

non-symmetric | SmallGroup(32,29) | 29 | Yes | |

non-symmetric | SmallGroup(32,30) | 30 | Yes | |

non-symmetric | SmallGroup(32,31) | 31 | Yes | |

non-symmetric | SmallGroup(32,32) | 32 | Yes | |

non-symmetric | SmallGroup(32,33) | 33 | Yes | |

non-symmetric | SmallGroup(32,34) | 34 | Yes | |

non-symmetric | SmallGroup(32,35) | 35 | Yes | |

non-symmetric | direct product of D8 and V4 | 46 | No | |

non-symmetric | direct product of Q8 and V4 | 47 | No | |

non-symmetric | direct product of SmallGroup(16,13) and Z2 | 48 | Yes |