Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C22×#1719 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(6,9)(7,10)(11,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,27)(26,28);(5,6,10,12,8,11,7,9)(14,18)(19,26)(20,25)(21,27);(6,11)(9,12)(13,15,17,23,16,22,24,28)(14,19,20,21,18,26,25,27)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264285);