Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#34179 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,5,6)(9,13)(11,14)(15,16,20,18)(17,21,25,23)(19,26,27,22)(24,29,30,28);(7,8,10,12)(9,11,13,14)(16,18)(21,23)(22,26)(28,29);(7,9)(8,11)(10,13)(12,14)(15,17,20,25)(16,21,18,23)(19,24,27,30)(22,28,26,29);(15,18,20,16)(17,23,25,21)(19,26,27,22)(24,29,30,28);(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532616);