C₂×#30847 #7502082

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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

3402 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)(13,17)(14,15)(16,18);(7,8)(9,10)(12,14)(13,15)(17,18);(4,6)(8,10)(11,13,16,15)(12,14,18,17);(4,6)(11,14,15,12,16,17,13,18)

maximal subgroup of	C₂×#7519107, C₂×#7515736, C₂×#7502663
maximal normal subgroup of	C₂×#7519107, C₂×#7515736, C₂×#7502663

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409278);