C₂×#24404 #6437990

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

7088 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 14 points:

(1,2);(7,8)(9,13,14,11)(10,12);(3,4)(5,6)(7,9)(8,11)(10,14)(12,13);(4,6)(7,8)(9,11)(10,12)(13,14);(7,9)(8,11)(10,14)(12,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345186);