C₂×#24380 #6437727

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5514 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344923);