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×#37404 |
| 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);(4,7)(8,10)(11,12,16,20)(13,14,18,17)(15,23,24,19)(21,25,26,22)(27,28)(29,36)(30,31)(32,40)(33,35)(34,41)(37,39)(38,42);(13,18)(14,17)(21,26)(22,25)(27,29,33,30)(28,31,35,36)(32,37,41,38)(34,39,40,42);(3,4,6,7)(5,8,9,10)(11,13,16,18)(12,17,20,14)(15,21,24,26)(19,25,23,22)(27,30,33,29)(28,31,35,36)(32,38,41,37)(34,39,40,42);(3,5,6,9)(4,8,7,10)(11,14,16,17)(12,18,20,13)(15,22,24,25)(19,26,23,21)(27,31)(28,29)(30,35)(32,39)(33,36)(34,37)(38,40)(41,42);(11,15,16,24)(12,19,20,23)(13,21,18,26)(14,22,17,25)(27,32,33,41)(28,34,35,40)(29,37,30,38)(31,39,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541991);