C₂×#30511 #7500934

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 ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4276 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

maximal subgroup of	C₂×#7519855, C₂×#7502928, C₂×#7514060
maximal normal subgroup of	C₂×#7519855, C₂×#7502928, C₂×#7514060

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408130);