C₄×#559 #34168

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

1282 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27156);