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 | |
4155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,21)(20,28)(22,25)(23,27)(24,29)(26,32)(30,31)(33,34,39,43)(35,40,45,47)(36,46,38,41)(37,42,48,44);(2,3)(7,9)(8,10)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,32)(28,30)(29,31)(33,35,39,45)(34,40,43,47)(36,42,38,44)(37,46,48,41);(2,3)(7,9)(8,10)(15,16)(20,29)(24,28)(26,31)(30,32)(33,36,39,38)(34,41,43,46)(35,44,45,42)(37,40,48,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,22,29)(18,24,25,28)(19,26,27,31)(21,30,23,32)(33,37,39,48)(34,42,43,44)(35,41,45,46)(36,47,38,40);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,22,23)(18,19,25,27)(20,30,29,32)(24,26,28,31)(33,34)(35,40)(36,41)(37,42)(38,46)(39,43)(44,48)(45,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,19)(18,21)(20,31)(22,27)(23,25)(24,32)(26,29)(28,30)(33,38,39,36)(34,41,43,46)(35,44,45,42)(37,47,48,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627818);