#9143385

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

4162 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)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26)(33,34,39,44)(35,45,48,40)(36,41,47,42)(37,43,46,38);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,35)(34,40)(36,46)(37,47)(38,42)(39,48)(41,43)(44,45);(2,4)(8,10)(9,12)(15,16)(18,20)(19,27)(25,30)(26,29)(33,36,39,47)(34,41,44,42)(35,46,48,37)(38,40,43,45);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,36)(34,42)(35,46)(37,48)(38,40)(39,47)(41,44)(43,45);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,37,39,46)(34,43,44,38)(35,47,48,36)(40,41,45,42);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,38,39,43)(34,37,44,46)(35,42,48,41)(36,40,47,45)

Polycyclic presentation:

2^9|a^2=gh,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, 9050581);