D₄₀ #678

order	80
isoclinism	isoclinic class of #677
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(1,ℤ/8) (order 640) · Inn(G) = D₂₀ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₄₀

Fitting	C₅×C₈ ⋊ C₂
Sylow	C₅ ⋊ D₈

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:

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

Polycyclic presentation:

2^4,5|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 7);