Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#49549 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8258 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);(4,6)(15,16,20,23)(17,24,19,21)(18,22,27,30)(25,29,26,28);(8,11)(9,13)(15,17,20,19)(16,21,23,24)(18,26,27,25)(22,29,30,28);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(15,18,20,27)(16,22,23,30)(17,25,19,26)(21,28,24,29);(7,8)(9,12)(10,11)(13,14)(15,19,20,17)(16,21,23,24)(18,25,27,26)(22,29,30,28);(7,9)(8,12)(10,13)(11,14)(15,16)(17,24)(18,22)(19,21)(20,23)(25,29)(26,28)(27,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612428);