C₂×#25014 #6443979

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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2642 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)(9,11)(10,12)(13,14);(3,4,6,5)(7,9,10,13)(8,11,12,14)(16,19)(17,21)(20,22);(8,12)(11,14)(15,16,18,20)(17,22,21,19);(3,5,6,4)

maximal subgroup of	C₂×#6444787, C₂×#6577868, C₂×D₄×#483, …
maximal normal subgroup of	C₂×#6444787, C₂×#6577868, C₂×D₄×#483, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351175);