Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51475 |
| 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 | |
4480 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);(5,9)(7,10)(12,16)(17,20)(18,23)(24,26)(27,28,32,33)(29,37,38,34)(30,35,36,31)(39,42,41,40);(3,4,6,8)(5,7,9,10)(12,16)(13,19)(14,22)(24,26)(28,33)(29,38)(31,35)(39,41);(3,5)(4,7)(6,9)(8,10)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29,32,38)(28,34,33,37)(30,41,36,39)(31,40,35,42);(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,36)(28,35,33,31)(29,39,38,41)(34,40,37,42);(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,31,32,35)(28,36,33,30)(29,40,38,42)(34,39,37,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614538);