C₅×D₈ #696

order	80
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₄×AGL(1,ℤ/8) (order 128) · Inn(G) = D₄ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

C₅×D₈

	subgroup	G/H
center, socle	C₂×C₅	D₄
commutator, Frattini, cosocle	C₄	C₂²×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ G

22 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)(3,5)(4,7)(6,8)(9,10,11,12,13);(2,5)(3,8)(6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 25);