#9143384

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

4082 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=h,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, 9050580);