D₄×#2669 #72335252

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

50347 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:

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

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,4)(14,16), (1,2)(3,4)(5,6,8,7)(10,12), (5,6)(7,8)(14,16), (6,7)(9,10,11,12)(13,14)(15,16), (9,10)(11,12), (1,2)(3,4)(5,7,8,6));