D₄×#449 #6573533

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#447
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5218 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480729);