C₂×#30509 #7500932

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄²

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

4600 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)(11,12,13,15)(14,16,18,17);(3,4)(5,6)(7,8)(9,10)(12,15)(16,17);(8,10)(12,16)(14,18)(15,17);(4,6)(12,17)(14,18)(15,16)

maximal subgroup of	C₂×#7519860, C₂×#7502919, C₂×#6676918, …
maximal normal subgroup of	C₂×#7519860, C₂×#7502919, C₂×#6676918, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408128);