C₂×#18256 #6346552

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

4950 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253748);