C₂×#25852 #6445802

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4168 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);(3,4,7,10)(5,9,6,8)(11,12)(13,15)(14,16)(17,18);(3,5,7,6)(4,8,10,9)(11,13)(12,15)(14,18)(16,17);(11,12)(13,17,18,15)(14,16);(3,6,7,5)(4,9,10,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352998);