#9143339

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

4059 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)(3,8)(4,9)(5,12)(6,7)(10,16)(11,14)(13,15)(17,18)(19,27)(20,25)(21,28)(22,23)(24,29)(26,31)(30,32);(1,2)(3,8)(4,13)(5,10)(6,7)(9,15)(11,14)(12,16)(18,23)(20,30)(24,27)(26,28)(33,34,38,42)(35,40,45,39)(36,37,41,47)(43,48,46,44);(2,7)(3,11)(9,13)(10,12)(17,19,22,29)(18,24,23,27)(20,28,30,26)(21,25,31,32)(33,35,38,45)(34,39,42,40)(36,43,41,46)(37,44,47,48);(1,3,6,11)(2,8,7,14)(4,10,15,12)(5,9,16,13)(17,19)(18,24)(20,26)(21,25)(22,29)(23,27)(28,30)(31,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,4)(2,9)(3,12)(5,8)(6,15)(7,13)(10,11)(14,16)(17,20,22,30)(18,25,23,32)(19,28,29,26)(21,24,31,27)(33,36)(34,37)(35,43)(38,41)(39,44)(40,48)(42,47)(45,46);(1,5)(2,10)(3,9)(4,14)(6,16)(7,12)(8,15)(11,13)(17,21,22,31)(18,26,23,28)(19,25,29,32)(20,27,30,24)(33,37)(34,41)(35,44)(36,42)(38,47)(39,46)(40,43)(45,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,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, 9050535);