Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#34048 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,23)(18,22)(19,21)(20,24)(25,29)(26,28)(27,30);(7,9,10,13)(8,11,12,14)(15,17)(16,20)(18,25)(19,24)(21,23)(22,27)(26,30)(28,29);(8,12)(11,14)(17,24)(20,23)(25,30)(27,29);(16,21)(20,23)(22,28)(27,29);(15,18,19,26)(16,22,21,28)(17,25,24,30)(20,27,23,29)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532476);