C₂×#3056 #60443

order	256
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1449 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(3,4)(5,6)(7,8,11,14)(9,15,16,22)(10,18,19,13)(12,20,21,17);(7,9,10,17,11,16,19,20)(8,12,13,15,14,21,18,22);(4,6);(9,16)(12,21)(15,22)(17,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53431);