# Presentations for groups of order 8

This article gives specific information, namely, presentations, about a family of groups, namely: groups of order 8.

View presentations for group families | View presentations for groups of a particular order |View other specific information about groups of order 8

## Power-commutator presentations

Each of the power-commutator presentations uses three generators (because that we call . The power relations are of the form where are natural numbers that depend on the nature of the group, and where depends on the nature of the group. The commutator relation is of the form where depends on the nature of the group. Note that , , and are always the identity.

The squaring comes because the underlying prime of the group is 2.

It turns out that for the isomorphism class of the final group, all the four values matter only mod the underlying prime, which in this case is 2. Thus, for simplicity, we assume that they are in the set . Note that this is a general feature of power-commutator presentations.

### Simplified power-commutator presentations

We here provide a *single* power-commutator presentation among the many possibilities.

Group | Second part of GAP ID (GAP ID is (p^3,2nd part) | Nilpotency class | Minimum size of generating set | Prime-base logarithm of exponent | full power-commutator presentation | ||||
---|---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 1 | 1 | 3 | 1 | 0 | 1 | 0 | [SHOW MORE] |

direct product of Z4 and Z2 | 2 | 1 | 2 | 2 | 0 | 1 | 0 | 0 | [SHOW MORE] |

dihedral group:D8 | 3 | 2 | 2 | 2 | 0 | 0 | 0 | 1 | [SHOW MORE] |

quaternion group | 4 | 2 | 2 | 2 | 0 | 1 | 1 | 1 | [SHOW MORE] |

elementary abelian group:E8 | 5 | 1 | 3 | 1 | 0 | 0 | 0 | 0 | [SHOW MORE] |

### Determining the isomorphism class from an arbitrary power-commutator presentation

Note that for the second and fourth groups, there are multiple sets of possible conditions, given in two separate rows within those groups. These are equivalent under permutations of the generators.

Note that because the prime is 2, and 0 and 1 are the only possibilities. Thus, saying "nonzero" means that the value must be 1.

Group | Second part of GAP ID (GAP ID is (p^3,2nd part) | Nilpotency class | Minimum size of generating set | Prime-base logarithm of exponent | Condition on | Condition on | Condition on | Condition on | Total number of power-commutator presentations |
---|---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 1 | 1 | 3 | nonzero | arbitrary | nonzero | zero | 2 |

direct product of Z4 and Z2 | 2 | 1 | 2 | 2 | arbitrary nonzero zero nonzero |
nonzero arbitrary zero zero |
zero zero nonzero nonzero |
zero zero zero zero |
5 |

dihedral group:D8 | 3 | 2 | 2 | 2 | arbitrary arbitrary |
zero arbitrary |
arbitrary zero |
nonzero | 3 |

quaternion group | 4 | 2 | 2 | 2 | zero | nonzero | nonzero | nonzero | 1 |

elementary abelian group:E8 | 5 | 1 | 3 | 1 | zero | zero | zero | zero | 1 |

not a group of order 8 (i.e., the presentation is not consistent) | -- | -- | -- | -- | nonzero | arbitrary | arbitrary | nonzero | 4 |

Total | -- | -- | -- | -- | -- | -- | -- | -- | 16 (equals ) |

Below, we list all the 16 possibilities for the 4-tuple and the corresponding group:

Group | GAP ID 2nd part | ||||
---|---|---|---|---|---|

0 | 0 | 0 | 0 | elementary abelian group:E8 | 5 |

1 | 0 | 0 | 0 | direct product of Z4 and Z2 | 2 |

0 | 1 | 0 | 0 | direct product of Z4 and Z2 | 2 |

1 | 1 | 0 | 0 | direct product of Z4 and Z2 | 2 |

0 | 0 | 1 | 0 | direct product of Z4 and Z2 | 2 |

1 | 0 | 1 | 0 | cyclic group:Z8 | 1 |

0 | 1 | 1 | 0 | direct product of Z4 and Z2 | 2 |

1 | 1 | 1 | 0 | cyclic group:Z8 | 1 |

0 | 0 | 0 | 1 | dihedral group:D8 | 3 |

1 | 0 | 0 | 1 | not a group of order 8 (actually becomes Klein four-group) | -- |

0 | 1 | 0 | 1 | dihedral group:D8 | 3 |

1 | 1 | 0 | 1 | not a group of order 8 | -- |

0 | 0 | 1 | 1 | dihedral group:D8 | 3 |

1 | 0 | 1 | 1 | not a group of order 8 | -- |

0 | 1 | 1 | 1 | quaternion group | 4 |

1 | 1 | 1 | 1 | not a group of order 8 | -- |