C₂×#30857 #7502097

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
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 ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3656 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)(12,14)(13,15)(17,18);(7,8)(9,10)(11,12)(13,14)(15,18)(16,17);(4,6)(8,10)(11,13,16,15)(12,14,17,18);(4,6)(11,14,13,12,16,18,15,17)

maximal subgroup of	C₂×#7519107, C₂×#7515965, C₂×#7502919
maximal normal subgroup of	C₂×#7519107, C₂×#7515965, C₂×#7502919

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409293);