Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105185 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #396 |
| commutator, Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,4,7,3,6)(10,13)(11,14)(15,16)(18,21)(22,23);(9,10,14,16,12,15,11,13)(17,18,19,21)(20,22,24,23);(1,2,4,7)(3,8,5,6)(9,10,12,15)(11,16,14,13)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=dehi,b^2=dg,c^2=h,d^2=g,e^2=h,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fi,d^c=dg,e^c=eh,f^c=fgi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12400);