#9066133

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

4146 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8973329);