order	248
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×C₅×S₄ (order 720) · Inn(G) = C₂² · Out(G) = C₂×C₃×C₅×D₃
homology	Abelianization (H₁) = C₂²×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

Q₈×C₃₁

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂×C₃₁	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₃₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ G

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 248

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(1,5,3,7)(2,6,4,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(248, 10);