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

Central product of D8 and Q8 | 32 (50) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (4) Subgroup rank of a group (4) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (1) |

Central product of D8 and Z4 | 16 (13) | 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) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (1) |

Central product of UT(3,Z) and Z identifying center with 2Z | | Nilpotency class (2) Derived length (2) Frattini length (2) Hirsch length of a polycyclic group (3) Polycyclic breadth (3) |

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: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: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: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) |

Dihedral group:D16 | 16 (7) | 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) Nilpotency class (3) Derived length (2) 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 (1) Characteristic rank of a p-group (1) Number of conjugacy classes (7) Number of equivalence classes under rational conjugacy (6) Number of conjugacy classes of rational elements (5) |

Dihedral group:D32 | 32 (18) | 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 (16) Prime-base logarithm of exponent (4) Nilpotency class (4) Derived length (2) Frattini length (4) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (1) Characteristic rank of a p-group (1) |

Dihedral group:D8 | 8 (3) | 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 (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Fitting 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 (1) Number of conjugacy classes (5) Number of equivalence classes under real conjugacy (5) Number of conjugacy classes of real elements (5) Number of equivalence classes under rational conjugacy (5) Number of conjugacy classes of rational elements (5) Number of subgroups (10) Number of conjugacy classes of subgroups (8) Number of normal subgroups (6) Number of automorphism classes of subgroups (6) Number of characteristic subgroups (4) |

Direct product of D16 and Z2 | 32 (39) | 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) Nilpotency class (3) Derived length (2) 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 (2) Characteristic rank of a p-group (2) |

Direct product of D8 and Z2 | 16 (11) | 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) Nilpotency class (2) Derived length (2) Frattini length (2) Fitting 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 (2) |

Direct product of D8 and Z3 | 24 (10) | Order of a group (24) Exponent of a group (12) Frattini length (2) Fitting length (1) Derived length (2) Nilpotency class (2) |

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 Q8 and Z2 | 16 (12) | 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) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

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 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) |

Direct product of prime-cube order group:U(3,3) and Z3 | 81 (12) | |

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: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) |

Faithful semidirect product of E8 and Z4 | 32 (6) | 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) Nilpotency class (3) Derived length (2) Frattini length (2) Minimum size of generating set (2) 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) |

Free nilpotent group | | |

Generalized dihedral group for direct product of Z4 and Z4 | 32 (34) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (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 (2) |

Generalized quaternion group:Q16 | 16 (9) | 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) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Number of conjugacy classes (7) Number of equivalence classes under rational conjugacy (6) Number of conjugacy classes of rational elements (5) |

Inner holomorph of D8 | 32 (49) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (4) Subgroup rank of a group (4) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (1) |

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) |

M16 | 16 (6) | 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) Nilpotency class (2) Derived length (2) 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) |

M27 | 27 (4) | Order of a group (27) Underlying prime of p-group (3) Prime-base logarithm of order (3) Exponent of a group (9) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (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) |

M64 | 64 (51) | 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 (32) Prime-base logarithm of exponent (5) Nilpotency class (2) Derived length (2) Frattini length (5) 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) |

M81 | 81 (6) | Order of a group (81) Exponent of a group (27) Frattini length (3) Derived length (2) Nilpotency class (2) Subgroup rank of a group (2) Minimum size of generating set (2) Rank of a p-group (2) Normal rank of a p-group (2) |

Nontrivial semidirect product of Z4 and Z4 | 16 (4) | 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) Nilpotency class (2) Derived length (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) Number of subgroups (15) Number of conjugacy classes of subgroups (13) Number of conjugacy classes (10) |

Quaternion group | 8 (4) | 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) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Number of conjugacy classes (5) Number of equivalence classes under real conjugacy (5) Number of conjugacy classes of real elements (5) Number of equivalence classes under rational conjugacy (5) Number of conjugacy classes of rational elements (5) Number of subgroups (6) Number of conjugacy classes of subgroups (6) Number of normal subgroups (6) Number of automorphism classes of subgroups (4) |

Semidihedral group:SD16 | 16 (8) | 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) Nilpotency class (3) Derived length (2) 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 (1) Characteristic rank of a p-group (1) Number of conjugacy classes (7) Number of equivalence classes under rational conjugacy (6) Number of conjugacy classes of rational elements (5) |

Semidihedral group:SD32 | 32 (19) | 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 (16) Prime-base logarithm of exponent (4) Nilpotency class (4) Derived length (2) Frattini length (4) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (1) Characteristic rank of a p-group (1) |

Semidirect product of Z8 and Z4 of M-type | 32 (4) | 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) Nilpotency class (2) Derived length (2) 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) |

SmallGroup(16,3) | 16 (3) | 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) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (2) 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) Number of conjugacy classes (10) Number of conjugacy classes of subgroups (17) |