Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36024 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,5)(8,11)(12,14)(15,16,20,18)(17,23,24,21)(19,22,27,26)(25,29,30,28);(3,4,6,5)(17,24)(21,23)(25,30)(28,29);(3,5,6,4)(7,8)(9,12)(10,11)(13,14)(15,17)(16,21)(18,23)(19,25)(20,24)(22,28)(26,29)(27,30);(7,9,10,13)(8,12,11,14)(15,18,20,16)(17,23,24,21)(19,22,27,26)(25,28,30,29);(15,19)(16,22)(17,25)(18,26)(20,27)(21,28)(23,29)(24,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536708);