#9224337

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

4209 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9131533);