Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,9)(5,7)(11,12,16,13)(14,19,21,17)(15,18,23,20)(22,25,26,24)(27,28)(29,37)(30,35)(31,36)(32,33)(34,40)(38,39)(41,42);(3,4,8,9)(5,6,7,10)(12,13)(14,21)(18,20)(22,26)(28,33)(30,36)(34,37)(38,42);(3,5,8,7)(4,10,9,6)(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,29,32,40)(28,34,33,37)(30,38,36,42)(31,39,35,41);(3,6,8,10)(4,5,9,7)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,30,32,36)(28,35,33,31)(29,38,40,42)(34,41,37,39);(3,7,8,5)(4,10,9,6)(11,15,16,23)(12,18,13,20)(14,22,21,26)(17,24,19,25)(27,31,32,35)(28,36,33,30)(29,39,40,41)(34,42,37,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999736);