D₄×#537 #7664859

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

11818 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)(11,15)(12,13)(14,16);(10,13)(14,15);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(9,11,12,16)(10,14,13,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572055);