#8399488

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

4130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306684);