C₂×#26041 #6443358

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

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

2522 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

maximal subgroup of	C₂×D₄×#478, C₂×#6578973, C₂×#6444783
maximal normal subgroup of	C₂×D₄×#478, C₂×#6578973, C₂×#6444783

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350554);