C₂×#30501 #7500923

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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

4314 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)(11,12,14,18)(13,17,15,16);(3,4)(5,6)(7,8)(9,10)(11,13,14,15)(12,16,18,17);(8,10)(12,16)(13,15)(17,18);(4,6)(12,17)(13,15)(16,18)

maximal subgroup of	C₂×#7519633, C₂×#6676918, C₂×#7502671, …
maximal normal subgroup of	C₂×#7519633, C₂×#6676918, C₂×#7502671, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408119);