Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #104897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
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 | |
285 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,5,9)(3,10,11,7)(4,6,13,8)(12,15,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,37,41)(35,38,43,48)(36,39,45,40)(42,46,44,47);(2,6)(3,11)(4,13)(7,15)(8,9)(10,14)(17,19,22,29)(18,23,26,32)(20,28,21,27)(24,31,25,30)(33,35,37,43)(34,38,41,48)(36,44,45,42)(39,47,40,46);(1,3,4,12,5,11,13,16)(2,7,8,15,9,10,6,14)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31)(34,39)(36,45)(38,46)(40,41)(42,44)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=d,d^2=g,e^2=h,f^2=hi,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=fhi,e^c=egh,f^c=fghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12109);