#9159939

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

4187 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18)(19,23)(20,25)(21,28)(22,24)(26,32)(27,30)(29,31)(33,34,39,43)(35,44,42,38)(36,47,46,40)(37,41,45,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,22,27)(18,23,24,30)(20,26,31,28)(21,25,32,29)(33,35,39,42)(34,38,43,44)(36,45,46,37)(40,48,47,41);(4,13)(8,12)(11,14)(15,16)(18,24)(19,27)(25,29)(26,28)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,48,44,41);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32)(33,36)(34,40)(35,45)(37,42)(38,48)(39,46)(41,44)(43,47);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,27)(30,32)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,47,44,40);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,21)(18,26)(19,25)(20,30)(22,32)(23,31)(24,28)(27,29)(33,38)(34,42)(35,43)(36,48)(37,40)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9067135);