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) = C₁

Aliases

C₅×Q₁₆

Subgroups

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

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,19,20,21);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 27);