#8633509

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540705);