AGL(1,𝔽₈)⋊C₉ #92626

order	504
isoclinism	isoclinic class of AΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×AΓL(1,𝔽₈) (order 504) · Inn(G) = AΓL(1,𝔽₈) · Out(G) = C₃
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

derived, Hall	AGL(1,𝔽₈) ⋊ C₉
Sylow	C₂³ ⋊ C₇⋊C₉

	subgroup	G/H
center, Frattini, hypercenter	C₃	AΓL(1,𝔽₈)
commutator, hypocenter	AGL(1,𝔽₈)	C₉
Fitting, socle	C₂³×C₃	ΓL(1,𝔽₈)
prime radical, cosocle	C₃×AGL(1,𝔽₈)	C₃
LCS	G ▷ AGL(1,𝔽₈)
UCS	C₁ ◁ C₃

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

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

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

Polycyclic presentation:

3^2,7,2^3|a^3=b,c^a=c^2,d^a=df,e^a=de,f^a=d,d^c=e,e^c=f,f^c=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 52);