Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#22055 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,6,5)(7,8)(9,13)(10,12)(11,14)(15,16)(17,27)(18,22)(19,26)(20,25)(21,24)(23,29)(28,30);(3,5,6,4)(7,9)(8,11)(10,14)(12,13)(15,17)(16,22)(18,20)(19,29)(21,28)(23,24)(25,27)(26,30);(9,14)(11,13)(15,18,20,17)(16,23,25,30)(19,29,21,28)(22,26,27,24);(8,12)(11,13)(15,19,20,21)(16,24,25,26)(17,28,18,29)(22,30,27,23)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282701);