Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C24×C43 |
| direct product | C2×#14601 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,6,5)(8,11)(9,13)(15,16,19,17)(18,21,22,20)(24,25)(28,29);(16,17)(18,22)(23,24,27,25)(26,29,30,28);(3,5,6,4)(7,8)(9,12)(10,11)(13,14)(15,17,19,16)(18,21,22,20)(23,25,27,24)(26,29,30,28);(7,9,10,13)(8,12,11,14)(15,18,19,22)(16,20,17,21)(23,26)(24,28)(25,29)(27,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428288);