C₂×#25012 #6443985

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#2537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2934 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)(15,16,18,21)(17,19,22,20);(8,12)(11,14)(16,19)(17,22)(20,21);(3,5,6,4)

maximal subgroup of	C₂×#6444784, C₂×#7520387, C₂×#6585523, …
maximal normal subgroup of	C₂×#6444784, C₂×#7520387, C₂×#6585523, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351181);