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×#37086 |
| 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 | |
4366 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);(4,7)(8,10)(14,21)(18,23)(20,25)(24,26)(27,28,32,31)(29,35,37,33)(30,38,39,34)(36,40,42,41);(12,16)(13,19)(18,23)(20,25)(27,29,32,37)(28,33,31,35)(30,36,39,42)(34,40,38,41);(3,4,6,7)(5,8,9,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,29)(28,33)(30,36)(31,35)(32,37)(34,40)(38,41)(39,42);(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,30,32,39)(28,34,31,38)(29,36,37,42)(33,40,35,41);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,28)(29,35,37,33)(30,38,39,34)(36,41,42,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541528);