C₂×#30855 #7502090

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3364 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)(11,12)(13,14)(15,16)(17,18);(7,8)(9,10)(12,13)(14,18)(16,17);(4,6)(8,10)(12,16)(13,17);(4,6)(11,13,14,12,15,17,18,16)

maximal subgroup of	C₂×#7515961, C₂×#7519100, C₂×#7502928
maximal normal subgroup of	C₂×#7515961, C₂×#7519100, C₂×#7502928

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409286);