Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #104917 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #394 |
| commutator, Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
296 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,13)(11,14)(15,16)(18,23)(19,26)(20,21)(24,27)(25,30)(28,29)(33,34,38,42)(35,36,40,46)(37,43,48,41)(39,47,45,44);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,18,22,25)(19,27,29,24)(20,30,21,23)(26,32,28,31)(33,35)(34,39)(36,43)(37,47)(38,40)(41,46)(42,45)(44,48);(1,3,4,10,5,11,12,16)(2,7,8,15,9,14,6,13)(17,19,20,28,22,29,21,26)(18,24,23,31,25,27,30,32)(33,36,37,45,38,46,48,39)(34,40,41,47,42,35,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=d,d^2=g,e^2=h,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fghi,e^c=egh,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12127);