C₂×#23587 #6432073

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4140 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);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,17)(16,18);(4,7)(8,10)(13,18)(15,16);(12,15)(13,18)(16,17);(3,5,6,9)(4,8,7,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339269);