Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×C4×#563 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7206 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);(3,4,5,6);(8,11)(9,13)(15,16,20,18)(17,23,25,21)(19,26,28,22)(24,29,27,30);(8,11)(12,14)(15,17,20,25)(16,21,18,23)(19,27,28,24)(22,30,26,29);(7,8)(9,12)(10,11)(13,14)(15,18,20,16)(17,21,25,23)(19,26,28,22)(24,29,27,30);(7,9)(8,12)(10,13)(11,14)(15,19)(16,22)(17,24)(18,26)(20,28)(21,29)(23,30)(25,27)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533131);