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×#51347 |
| 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 | |
4398 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);(5,9)(7,10)(27,28,32,30)(29,35,37,33)(31,38,39,34)(36,40,42,41);(3,4,6,8)(5,7,9,10)(12,16)(13,19)(14,22)(24,26)(28,30)(33,35)(34,38)(40,41);(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)(28,33)(30,35)(31,36)(32,37)(34,40)(38,41)(39,42);(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,28)(29,35,37,33)(31,38,39,34)(36,41,42,40);(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,31,32,39)(28,34,30,38)(29,36,37,42)(33,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614101);