Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37062 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,6,8)(5,7,9,10)(14,21)(18,23)(20,25)(24,26)(28,30)(29,35)(34,38)(37,41);(12,16)(13,19)(18,23)(20,25)(27,28,32,30)(29,36,35,33)(31,34,39,38)(37,42,41,40);(3,4)(5,7)(6,8)(9,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,29,32,35)(28,33,30,36)(31,37,39,41)(34,40,38,42);(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,30,32,28)(29,36,35,33)(31,38,39,34)(37,42,41,40);(3,5,6,9)(4,7,8,10)(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,39)(28,34,30,38)(29,37,35,41)(33,40,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541527);