Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2606 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)(5,6)(7,8)(9,11)(10,12)(13,14)(16,20)(17,23)(18,25)(21,26)(22,28)(24,29);(7,9,10,13)(8,11,12,14)(15,16,18,20,19,21,25,26)(17,22,23,27,24,28,29,30);(8,12)(11,14)(16,21)(20,26)(22,28)(27,30);(4,6)(15,17,18,23,19,24,25,29)(16,22,20,27,21,28,26,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536918);