Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#23681 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C22×D4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2562 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)(9,12)(10,11)(13,14)(15,16)(17,20)(18,21)(19,22)(23,30)(24,28)(25,27)(26,29);(8,11)(12,14)(15,17,18,24)(16,20,21,28)(19,23,26,25)(22,30,29,27);(4,6)(16,21)(17,23)(19,26)(20,27)(24,25)(28,30);(7,9,10,13)(8,12,11,14)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339449);