Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C4×#401 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle, prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3594 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);(3,4,5,6);(7,8,11,13)(9,14,10,12)(15,16,18,20)(17,19,21,22)(23,24)(25,27)(26,28)(29,30);(7,9,11,10)(8,12,13,14)(15,17,18,21)(16,19,20,22)(23,25,26,29)(24,27,28,30);(7,10,11,9)(8,12,13,14)(16,20)(19,22)(24,28)(25,29)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421240);