C₂×#24176 #6448753

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

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

4020 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)(5,9)(6,7)(8,10)(11,12)(13,15)(14,17)(16,18);(4,7)(8,9)(13,18)(15,16);(12,15)(13,18)(16,17);(3,5,6,10)(4,8,7,9)

maximal subgroup of	C₂×#248071, C₂×#6453829
maximal normal subgroup of	C₂×#248071, C₂×#6453829

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355949);