Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#37375 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8714 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);(4,6)(8,11)(9,13)(15,16,20,22)(17,23,24,18)(19,27,28,21)(25,26,30,29);(7,8,10,11)(9,12,13,14)(17,24)(18,23)(25,30)(26,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,25)(20,24)(21,26)(22,23)(27,29)(28,30);(7,9)(8,12)(10,13)(11,14)(15,18)(16,17)(19,26)(20,23)(21,25)(22,24)(27,30)(28,29);(15,19,20,28)(16,21,22,27)(17,25,24,30)(18,26,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541982);