C₂×#30499 #7500921

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×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

4638 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)(14,17)(16,18);(3,4)(5,6)(7,8)(9,10);(8,10)(12,15)(13,18)(16,17);(4,6)(12,15)(13,18)(16,17)

maximal subgroup of	C₂×#7519860, C₂×#7502663, C₂×#7502680, …
maximal normal subgroup of	C₂×#7519860, C₂×#7502663, C₂×#7502680, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408117);