C₂×#18624 #6348199

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3768 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);(4,8)(5,6)(9,10)(12,15)(13,17)(16,18);(3,4)(5,8)(6,9)(7,10)(13,17)(15,18);(3,5,7,6)(4,8,10,9)(11,12)(13,18)(14,16)(15,17);(3,5,7,6)(4,9,10,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255395);