C₅×Q₈ #178

order	40
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₄×S₄ (order 96) · Inn(G) = C₂² · Out(G) = C₄×D₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×#5935
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

C₅×Q₈

	subgroup	G/H
center, socle, prime radical	C₂×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ G

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 11);