#8720834

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628030);