D₄×#535 #7664858

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

13912 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572054);