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 | |
4238 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)(3,7)(4,8)(5,13)(6,10)(9,14)(11,16)(12,15)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(33,34,40,39)(35,43,37,41)(36,42,46,48)(38,44,45,47);(1,3,6,4)(2,7,10,8)(5,11,14,12)(9,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,35,40,37)(34,41,39,43)(36,45,46,38)(42,44,48,47);(1,2,6,10)(3,8,4,7)(5,13,14,9)(11,15,12,16)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,36,40,46)(34,42,39,48)(35,45,37,38)(41,44,43,47);(1,4,6,3)(2,8,10,7)(5,12,14,11)(9,16,13,15)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,37,40,35)(34,43,39,41)(36,45,46,38)(42,44,48,47);(1,2)(3,8)(4,7)(5,9)(6,10)(11,16)(12,15)(13,14)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,38,40,45)(34,44,39,47)(35,36,37,46)(41,48,43,42);(1,5,6,14)(2,9,10,13)(3,11,4,12)(7,15,8,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,39,40,34)(35,43,37,41)(36,42,46,48)(38,47,45,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900447);