Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C4×#510 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C22×D8 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6);(7,8)(9,17)(10,21)(11,26)(12,14)(13,18)(15,24)(16,23)(19,37)(20,34)(22,32)(25,33)(27,31)(28,36)(29,38)(30,35);(7,9,12,20)(8,13,14,28)(10,22,25,38)(11,27,23,19)(15,31,30,37)(16,32,26,29)(17,33,34,21)(18,35,36,24);(8,14)(13,28)(15,30)(16,26)(17,34)(21,33)(29,32)(31,37)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529797);