#9159934

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

4094 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=hi,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, 9067130);