Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#58914 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4436 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);(3,4,7,9)(5,6,10,8)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(27,28,32,35)(29,33,36,31)(30,34,40,42)(37,41,38,39);(3,5,7,10)(4,8,9,6)(12,14)(17,19)(18,22)(24,25)(29,36)(31,33)(37,38)(39,41);(4,9)(6,8)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,29)(28,33)(30,38)(31,35)(32,36)(34,39)(37,40)(41,42);(3,6,7,8)(4,5,9,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30)(28,34)(29,37)(31,39)(32,40)(33,41)(35,42)(36,38);(3,5)(4,8)(6,9)(7,10)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,31,32,33)(28,29,35,36)(30,39,40,41)(34,37,42,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012514);