Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105185 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #396 |
| commutator, Frattini, cosocle | C43 | C23 |
| prime radical | #2867 | C22 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,7,8,5)(9,10,14,16)(11,13,15,12);(9,11,12,10,14,15,13,16)(17,18)(19,21)(20,23)(22,24);(2,5)(3,8)(6,7)(10,11)(12,13)(15,16)(18,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=de,b^2=d,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, 12381);