C₂³×#498 #10575263

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

9058 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482459);