Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,14,8)(11,16,13,15)(17,18)(19,20)(21,29)(22,24)(23,31)(25,27)(26,30)(28,32)(33,34,40,45)(35,39,44,41)(36,42,37,46)(38,43,47,48);(2,6)(3,9)(8,12)(11,13)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,30)(31,32)(33,35,40,44)(34,41,45,39)(36,38,37,47)(42,48,46,43);(2,6)(7,10)(8,12)(15,16)(17,18,22,24)(19,25,27,20)(21,23,31,29)(26,30,32,28)(33,36,40,37)(34,42,45,46)(35,38,44,47)(39,43,41,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,14)(13,15)(17,18)(19,25)(20,27)(21,23)(22,24)(26,30)(28,32)(29,31)(33,37,40,36)(34,42,45,46)(35,38,44,47)(39,48,41,43);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,20,22,25)(18,19,24,27)(21,30,31,28)(23,32,29,26)(33,38)(34,43)(35,37)(36,44)(39,46)(40,47)(41,42)(45,48);(1,4,5,14)(2,8,6,12)(3,11,9,13)(7,15,10,16)(17,21)(18,23)(19,26)(20,28)(22,31)(24,29)(25,30)(27,32)(33,39,40,41)(34,44,45,35)(36,43,37,48)(38,42,47,46)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154504);