C₂²×#3358 #10576468

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

9054 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10483664);