C₂×#13305 #498788

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#466	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1792 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,6)(11,12,15,19)(13,16,22,25)(14,23,21,18)(17,26,24,20);(7,8)(9,10)(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24);(4,6)(8,10)

maximal subgroup of	C₂×#498795, C₂×#499724, C₂×#499713
maximal normal subgroup of	C₂×#498795, C₂×#499724, C₂×#499713

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405984);