D₄×#547 #7699469

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

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

11765 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606665);