#9159950

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

4025 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)(3,7)(4,9)(5,12)(6,8)(10,16)(11,14)(13,15)(17,18)(19,29)(20,23)(21,30)(22,26)(24,28)(25,32)(27,31)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,3,6,11)(2,7,8,14)(4,10,15,12)(5,9,16,13)(17,19)(18,25)(20,30)(21,26)(22,27)(23,31)(24,32)(28,29)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,8)(3,11)(9,13)(10,12)(17,20,24,22)(18,26,28,23)(19,30,32,27)(21,29,31,25)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,2)(3,7)(4,13)(5,10)(6,8)(9,15)(11,14)(12,16)(17,21,24,31)(18,27,28,30)(19,26,32,23)(20,25,22,29)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,11)(14,16)(17,22,24,20)(18,26,28,23)(19,30,32,27)(21,25,31,29)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,5)(2,10)(3,9)(4,14)(6,16)(7,15)(8,12)(11,13)(17,23,24,26)(18,22,28,20)(19,31,32,21)(25,27,29,30)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=hi,e^2=g,f^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, 9067146);