#9143392

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

4032 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=gh,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, 9050588);