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×#34328 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,6,5)(7,8,11,10)(9,13,14,12)(16,17)(19,27)(21,23)(25,30);(15,16,20,17)(18,23,26,21)(19,22,27,24)(25,29,30,28);(7,9,11,14)(8,12,10,13)(15,17,20,16)(18,23,26,21)(19,24,27,22)(25,29,30,28);(7,8,11,10)(9,12,14,13)(15,18)(16,21)(17,23)(19,25)(20,26)(22,28)(24,29)(27,30);(3,5,6,4)(7,10,11,8)(9,13,14,12)(15,19)(16,22)(17,24)(18,25)(20,27)(21,28)(23,29)(26,30)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532804);