Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34348 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,5,6)(7,8)(9,15)(10,14)(11,13)(12,16)(17,21)(18,20)(19,22)(23,24)(25,28)(26,33)(27,30)(29,35)(31,38)(32,37)(34,36);(7,9)(8,12)(10,17)(11,16)(13,15)(14,19)(18,22)(20,21)(23,25,27,32)(24,28,30,37)(26,34,35,31)(29,38,33,36);(9,16)(12,15)(17,22)(19,21)(23,26,27,35)(24,29,30,33)(25,31,32,34)(28,36,37,38);(8,13)(9,16)(14,20)(17,22)(24,30)(28,37)(29,33)(36,38);(7,10,11,18)(8,14,13,20)(9,17,16,22)(12,19,15,21)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533126);