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Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
Abelian Cyclic Solvable Simple
Perfect Direct product Semidirect product Wreath product
Automorphism group Automorphisms Min. transitive degree Min. permutation degree
Number of subgroups Num. of normal subs Num. of conj. classes
Order statistics Family Order factorization Commutator length
Min. degree of $\C$-irrep Min. degree of $\Q$-irrep Min. degree of $\C$-rep Min. degree of $\Q$-rep
Outer aut. group Outer automorphisms Metabelian Metacyclic
Almost simple Quasisimple A-group Z-group
Frattini subgroup Derived length Rank Schur multiplier
Supersolvable Monomial Rational
Num. of char. subgroups Num. of aut. classes Number of divisions
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Results (10 matches)

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Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
1.1 $C_1$ $1$ $1$ 1 $C_1$ $C_1$ $C_1$ $C_1$ Trivial
2.1 $C_2$ $2$ $2$ 2 $C_2$ $C_1$ $C_1$ $C_{2}$ Cyclic
4.1 $C_4$ $2^{2}$ $2^{2}$ 4 $C_4$ $C_1$ $C_1$ $C_{4}$ Cyclic
8.4 $Q_8$ $2^{3}$ $2^{2}$ 5 $C_2$ $C_2^2$ $C_2$ $C_{2}^{2}$ Nilpotent - 2
16.9 $Q_{16}$ $2^{4}$ $2^{3}$ 7 $C_2$ $D_4$ $C_4$ $C_{2}^{2}$ Nilpotent - 3
32.20 $Q_{32}$ $2^{5}$ $2^{4}$ 11 $C_2$ $D_8$ $C_8$ $C_{2}^{2}$ Nilpotent - 4
64.54 $Q_{64}$ $2^{6}$ $2^{5}$ 19 $C_2$ $D_{16}$ $C_{16}$ $C_{2}^{2}$ Nilpotent - 5
128.163 $Q_{128}$ $2^{7}$ $2^{6}$ 35 $C_2$ $D_{32}$ $C_{32}$ $C_{2}^{2}$ Nilpotent - 6
256.541 $Q_{256}$ $2^{8}$ $2^{7}$ 67 $C_2$ $D_{64}$ $C_{64}$ $C_{2}^{2}$ Nilpotent - 7
512.2044 $Q_{512}$ $2^{9}$ $2^{8}$ 131 $C_2$ $D_{128}$ $C_{128}$ $C_{2}^{2}$ Nilpotent - 8
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