Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42689 |
| 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 | |
4414 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);(5,9)(7,10)(11,12,16,19)(13,17,20,15)(14,18,24,26)(21,25,22,23)(27,28)(29,37)(30,39)(31,36)(32,33)(34,40)(35,41)(38,42);(3,4,6,8)(5,7,9,10)(13,20)(15,17)(21,22)(23,25)(28,33)(31,42)(34,37)(35,39);(3,5,6,9)(4,7,8,10)(11,13)(12,17)(14,22)(15,19)(16,20)(18,23)(21,24)(25,26)(27,29,32,40)(28,34,33,37)(30,36,41,38)(31,39,42,35);(11,14)(12,18)(13,21)(15,23)(16,24)(17,25)(19,26)(20,22)(27,30,32,41)(28,35,33,39)(29,38,40,36)(31,34,42,37);(11,15,16,17)(12,13,19,20)(14,23,24,25)(18,21,26,22)(27,31,32,42)(28,36,33,38)(29,39,40,35)(30,34,41,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548500);