#8875557

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#48723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(3,9)(7,10)(11,15)(14,16)(17,18)(19,24)(20,30)(21,26)(22,23)(25,29)(27,32)(28,31)(34,39)(36,41)(37,45)(42,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(18,23)(21,30)(24,25)(28,32)(34,39)(36,41)(37,45)(42,48);(2,6)(7,10)(8,12)(14,16)(17,19,22,29)(18,24,23,25)(20,27,26,31)(21,28,30,32)(33,34,38,39)(35,41,43,36)(37,40,45,46)(42,47,48,44);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,25)(20,31)(21,28)(22,29)(23,24)(26,27)(30,32)(33,35,38,43)(34,36,39,41)(37,42,45,48)(40,44,46,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,26)(18,21,23,30)(19,27,29,31)(24,28,25,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,21)(18,26)(19,28)(20,23)(22,30)(24,31)(25,27)(29,32)(33,37,38,45)(34,40,39,46)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8782753);