C₅⋊#857 #91182

order	480
isoclinism	isoclinic class of C₅⋊#857
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅×C₄²⋊C₃	C₂
Frattini	C₂²	C₅⋊S₄
Fitting	C₄²×C₅	D₃
socle	C₂²×C₅	S₄
LCS	G ▷ C₅×C₄²⋊C₃
UCS	C₁

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

2,3,5,2^4|d^2=fg,e^2=f,b^a=b^2,c^a=c^4,d^a=e,e^a=d,f^a=fg,d^b=e,e^b=deg,f^b=g,g^b=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 258);