C₂×#9930 #234811

order	512
nilpotency class	3
isoclinism	isoclinic class of #9930
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2612 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2);(4,8)(5,6)(9,10)(11,12)(13,15)(14,16)(17,18);(3,4)(5,10)(6,8)(7,9)(15,18);(3,4)(5,8)(6,10)(7,9)(11,13)(12,15)(14,17)(16,18)

maximal subgroup of	#72231590, C₂×#234827
maximal normal subgroup of	#72231590, C₂×#234827

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142007);