C₂×#28677 #6466222

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2206 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373418);