C₂×#31057 #7503320

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

4602 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)(11,12,14,17,15,16,13,18);(7,8)(9,10)(11,12)(13,17)(14,18)(15,16);(4,6)(8,10)(12,16)(17,18);(4,6)(11,13)(14,15)(17,18)

maximal subgroup of	C₂×#7520325, C₂×#6676918
maximal normal subgroup of	C₂×#7520325, C₂×#6676918

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410516);