Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#14897 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3326 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);(4,6)(9,13)(12,14)(15,16,19,20)(17,18,22,21)(23,24)(25,26)(27,29)(28,30);(8,11)(12,14)(16,20)(17,22)(23,25,27,30)(24,26,29,28);(7,8)(9,12)(10,11)(13,14)(15,17,19,22)(16,21,20,18)(23,25)(24,28)(26,29)(27,30);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,18,19,21)(16,22,20,17)(23,26,27,28)(24,25,29,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515089);