# Maximal class group

From Groupprops

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter

View other prime-parametrized group properties | View other group properties

## Contents

## Definition

A **maximal class group** is a group of prime power order that has nilpotency class , and order , where is a prime number and . Equivalently, it has the following properties:

- The abelianization of the group, i.e., the quotient of the group by its commutator subgroup has order .
- The upper central series and lower central series coincide, and all the successive quotients (except the top-most quotient) are of order .

## Facts

### For 2-groups

- Classification of finite 2-groups of maximal class: There are only three possibilities in general for any order -- the dihedral group, semidihedral group, and generalized quaternion group. For order 8, two of these collapse into one and we get only two possibilities.
- Finite non-abelian 2-group has maximal class iff its abelianization has order four: In particular, all the finite 2-groups of a given order and maximal class are isoclinic groups.

### General facts

- P-group with derived subgroup of prime-square index not implies maximal class for odd p
- Group of exponent p and order greater than p^p is not embeddable in a maximal class group
- P-group is either absolutely regular or maximal class or has normal subgroup of exponent p and order p^p

## Examples

Prime number | Nilpotency class | Number of maximal class groups | List of maximal class groups | List of GAP IDs second part (in order of list) | |
---|---|---|---|---|---|

2 | 2 | 8 | 2 | dihedral group:D8, quaternion group | 3,4 |

2 | 3 | 16 | 3 | dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 | 7,8,9 |

2 | 4 | 32 | 3 | dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32 | 18,19,20 |

2 | 5 | 64 | 3 | dihedral group:D64, semidihedral group:SD64, generalized quaternion group:Q64 | 52,53,54 |

3 | 2 | 27 | 2 | prime-cube order group:U(3,3), M27 | 3,4 |

3 | 3 | 81 | 4 | wreath product of Z3 and Z3, SmallGroup(81,8), SmallGroup(81,9), SmallGroup(81,10) | 7,8,9,10 |

3 | 4 | 243 | 6 | ||

odd | 2 | 2 | prime-cube order group:U(3,p), semidirect product of cyclic group of prime-square order and cyclic group of prime order | 3,4 | |

odd | 3 | 4 | 7,8,9,10 | ||

odd | 4 |