C₂×#25663 #6450025

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	C₂×#3505	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4298 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);(3,4,6,7)(5,9,10,8)(11,12)(13,15)(14,16)(17,18);(4,7)(8,9)(11,13)(12,15)(14,18)(16,17);(11,12)(13,17,18,15)(14,16);(3,5,6,10)(4,8,7,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357221);