Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7462 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);(3,4,6,5)(8,11)(12,14)(15,16)(17,22)(18,23)(19,27)(20,21)(24,28)(25,26)(29,30);(3,5,6,4)(16,21)(18,22)(24,27)(26,29);(7,8)(9,12)(10,11)(13,14)(15,17,20,23)(16,22,21,18)(19,25,28,30)(24,29,27,26);(15,18,20,22)(16,23,21,17)(19,26,28,29)(24,30,27,25);(7,9,10,13)(8,12,11,14)(15,19)(16,24)(17,25)(18,26)(20,28)(21,27)(22,29)(23,30)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532697);