D₄×#131 #60145

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

1407 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2)(3,4);(2,4);(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53133);