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×#58648 |
| 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 | |
4468 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)(5,9)(6,7)(8,10)(27,28,31,32)(29,35,36,33)(30,38,39,34)(37,40,42,41);(4,7)(8,9)(12,16)(14,22)(17,19)(20,26)(28,32)(29,36)(34,38)(37,42);(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39)(41,42);(3,4)(5,8)(6,7)(9,10)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,29,31,36)(28,33,32,35)(30,37,39,42)(34,40,38,41);(3,5)(4,8)(6,10)(7,9)(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,30,31,39)(28,34,32,38)(29,37,36,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012021);