Additive group of p-adic integers | | |

Baer-Specker group | | Frattini length (1) Fitting length (1) |

Circle group | | Dimension of a real Lie group (1) Dimension of an algebraic group (1) |

Cyclic group:Z16 | 16 (1) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (16) Prime-base logarithm of exponent (4) Frattini length (4) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z2 | 2 (1) | Underlying prime of p-group (2) Order of a group (2) Prime-base logarithm of order (1) Max-length of a group (1) Chief length (1) Composition length (1) Exponent of a group (2) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) Number of conjugacy classes (2) Number of subgroups (2) Number of conjugacy classes of subgroups (2) Number of equivalence classes under real conjugacy (2) Number of equivalence classes under rational conjugacy (2) |

Cyclic group:Z20 | 20 (2) | Order of a group (20) Exponent of a group (20) Frattini length (2) Fitting length (1) Derived length (1) |

Cyclic group:Z27 | 27 (1) | Underlying prime of p-group (3) Order of a group (27) Prime-base logarithm of order (3) Max-length of a group (3) Chief length (3) Composition length (3) Exponent of a group (27) Prime-base logarithm of exponent (3) Frattini length (3) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z3 | 3 (1) | Underlying prime of p-group (3) Order of a group (3) Prime-base logarithm of order (1) Max-length of a group (1) Chief length (1) Composition length (1) Exponent of a group (3) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z32 | 32 (1) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (32) Prime-base logarithm of exponent (5) Frattini length (5) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z4 | 4 (1) | Underlying prime of p-group (2) Order of a group (4) Prime-base logarithm of order (2) Max-length of a group (2) Chief length (2) Composition length (2) Exponent of a group (4) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z5 | 5 (1) | Order of a group (5) Exponent of a group (5) Derived length (1) Frattini length (1) Fitting length (1) |

Cyclic group:Z64 | 64 (1) | Underlying prime of p-group (2) Order of a group (64) Prime-base logarithm of order (6) Max-length of a group (6) Chief length (6) Composition length (6) Exponent of a group (64) Prime-base logarithm of exponent (6) Frattini length (6) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z7 | 7 (1) | Order of a group (7) Exponent of a group (7) Frattini length (1) Fitting length (1) Subgroup rank of a group (1) Rank of a p-group (1) |

Cyclic group:Z8 | 8 (1) | Underlying prime of p-group (2) Order of a group (8) Prime-base logarithm of order (3) Max-length of a group (3) Chief length (3) Composition length (3) Exponent of a group (8) Prime-base logarithm of exponent (3) Frattini length (3) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z81 | 81 (1) | Underlying prime of p-group (3) Order of a group (81) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (81) Prime-base logarithm of exponent (4) Frattini length (4) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z9 | 9 (1) | Order of a group (9) Exponent of a group (9) Derived length (1) Frattini length (2) Fitting length (1) Subgroup rank of a group (1) |

Direct product of E8 and Z4 | 32 (45) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (4) Subgroup rank of a group (4) Rank of a p-group (4) Normal rank of a p-group (4) Characteristic rank of a p-group (4) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z16 and V4 | 64 (183) | Underlying prime of p-group (2) Order of a group (64) Prime-base logarithm of order (6) Max-length of a group (6) Chief length (6) Composition length (6) Exponent of a group (16) Prime-base logarithm of exponent (4) Frattini length (4) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z27 and Z3 | 81 (5) | Underlying prime of p-group (3) Order of a group (81) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (27) Prime-base logarithm of exponent (3) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z4 and Z2 | 8 (2) | Underlying prime of p-group (2) Order of a group (8) Prime-base logarithm of order (3) Exponent of a group (4) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z4 and Z4 | 16 (2) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (4) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z8 and V4 | 32 (36) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Frattini length (3) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z8 and Z2 | 16 (5) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (8) Prime-base logarithm of exponent (3) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z8 and Z4 | 32 (3) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z8 and Z8 | 64 (2) | Underlying prime of p-group (2) Order of a group (64) Prime-base logarithm of order (6) Max-length of a group (6) Chief length (6) Composition length (6) Exponent of a group (8) Prime-base logarithm of exponent (3) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z9 and E9 | 81 (11) | Underlying prime of p-group (3) Order of a group (81) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (9) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) Nilpotency class (1) Derived length (1) Fitting length (1) |

Direct product of Z9 and Z9 | 81 (2) | Underlying prime of p-group (3) Order of a group (81) Prime-base logarithm of order (4) Exponent of a group (9) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Elementary abelian group:E64 | 64 (267) | Underlying prime of p-group (2) Order of a group (64) Prime-base logarithm of order (6) Max-length of a group (6) Chief length (6) Composition length (6) Exponent of a group (2) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (6) Subgroup rank of a group (6) Rank of a p-group (6) Normal rank of a p-group (6) Characteristic rank of a p-group (6) Nilpotency class (1) Derived length (1) Fitting length (1) |

Elementary abelian group:E8 | 8 (5) | Underlying prime of p-group (2) Order of a group (8) Prime-base logarithm of order (3) Max-length of a group (3) Chief length (3) Composition length (3) Exponent of a group (2) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) Nilpotency class (1) Derived length (1) Fitting length (1) |

Elementary abelian group:E81 | 81 (15) | Underlying prime of p-group (3) Order of a group (81) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (3) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (4) Subgroup rank of a group (4) Rank of a p-group (4) Normal rank of a p-group (4) Characteristic rank of a p-group (4) Nilpotency class (1) Derived length (1) Fitting length (1) |

Elementary abelian group:E9 | 9 (2) | Underlying prime of p-group (3) Order of a group (9) Prime-base logarithm of order (2) Max-length of a group (2) Chief length (2) Composition length (2) Exponent of a group (3) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Free abelian group of rank two | | |

Group of integers | | Derived length (1) Nilpotency class (1) Fitting length (1) Frattini length (1) Subgroup rank of a group (1) |

Group of rational numbers modulo integers | | |

Group of rational numbers with square-free denominators | | |

Klein four-group | 4 (2) | Underlying prime of p-group (2) Order of a group (4) Prime-base logarithm of order (2) Max-length of a group (2) Chief length (2) Composition length (2) Exponent of a group (2) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Profinite completion of the integers | | |

Quasicyclic group | | |

Quotient of profinite completion of the integers by the integers | | |

Trivial group | 1 (1) | Order of a group (1) Exponent of a group (1) Nilpotency class (0) Derived length (0) Frattini length (0) Minimum size of generating set (0) Subgroup rank of a group (0) |