#8533419

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

3970 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:

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

Polycyclic presentation:

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

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, 8440615);