Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(11,15)(13,16)(17,18)(19,24)(20,22)(21,23);(9,10,12,14)(11,16,15,13)(17,19,22,21)(18,23,20,24);(2,4)(9,11,12,15)(10,13,14,16)(17,20,22,18)(19,23,21,24);(5,6,7,8)(17,19,22,21)(18,23,20,24);(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519498);