C₂×#28231 #6465414

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

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

3410 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

maximal subgroup of	C₂×#6465861, C₂×#6585523
maximal normal subgroup of	C₂×#6465861, C₂×#6585523

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372610);