Q₈×#117 #34124

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

1217 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27112);