#9224339

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

4205 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)(3,8)(4,14)(5,6)(7,10)(9,13)(11,16)(12,15)(17,18,22,26)(19,27,28,23)(20,24,31,30)(21,32,29,25)(34,40)(36,43)(41,42)(44,46);(1,3)(2,8)(4,15)(5,11)(6,16)(7,13)(9,10)(12,14)(17,19,22,28)(18,23,26,27)(20,29,31,21)(24,32,30,25)(33,34,39,40)(35,41,37,42)(36,38,43,48)(44,45,46,47);(1,4,7,5)(2,6,10,14)(3,11,13,15)(8,12,9,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31)(33,35,39,37)(34,41,40,42)(36,44,43,46)(38,45,48,47);(1,5,7,4)(2,6,10,14)(3,11,13,15)(8,16,9,12)(17,18)(19,23)(20,30)(21,32)(22,26)(24,31)(25,29)(27,28)(33,36)(34,38)(35,44)(37,46)(39,43)(40,48)(41,45)(42,47);(1,6)(2,5)(3,12)(4,10)(7,14)(8,15)(9,11)(13,16)(17,20,22,31)(18,24,26,30)(19,21,28,29)(23,25,27,32)(33,37,39,35)(34,42,40,41)(36,44,43,46)(38,45,48,47);(1,3)(2,9)(4,11)(5,15)(6,16)(7,13)(8,10)(12,14)(17,21)(18,25)(19,20)(22,29)(23,24)(26,32)(27,30)(28,31)(33,38)(34,43)(35,45)(36,40)(37,47)(39,48)(41,46)(42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,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, 9131535);