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×#55485 |
| 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 | |
4474 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);(12,16)(13,19)(18,23)(20,25)(27,28,31,32)(29,35,37,33)(30,38,39,34)(36,41,40,42);(3,4,7,8)(5,6,9,10)(12,16)(17,22)(18,23)(24,26)(28,32)(30,39)(33,35)(36,40);(4,8)(5,9)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,29)(28,33)(30,40)(31,37)(32,35)(34,42)(36,39)(38,41);(3,5)(4,6)(7,9)(8,10)(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,30,31,39)(28,34,32,38)(29,36,37,40)(33,41,35,42);(3,6)(4,5)(7,10)(8,9)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,28)(29,33)(30,34)(31,32)(35,37)(36,41)(38,39)(40,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009451);