#9159905

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

4259 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)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(34,39)(35,41)(37,45)(44,46);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,34,38,39)(35,40,41,36)(37,42,45,47)(43,46,48,44);(4,13)(8,12)(11,14)(15,16)(18,23)(19,27)(25,28)(26,29)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,46,47,44);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,35)(34,40)(36,39)(37,43)(38,41)(42,46)(44,47)(45,48);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,36,38,40)(34,41,39,35)(37,46,45,44)(42,43,47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=i,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, 9067101);