Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
| direct product | C2×#53307 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(8,10)(11,12,16,14)(13,19,21,17)(15,18,23,22)(20,25,26,24)(27,28)(29,36)(30,33)(31,39)(32,35)(34,40)(37,41)(38,42);(4,7)(8,10)(15,23)(18,22)(20,26)(24,25)(27,29,32,30)(28,33,35,36)(31,37,40,38)(34,41,39,42);(3,4)(5,8)(6,7)(9,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,30,32,29)(28,33,35,36)(31,38,40,37)(34,41,39,42);(3,5)(4,8)(6,9)(7,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,28)(29,36)(30,33)(31,34)(32,35)(37,42)(38,41)(39,40);(3,4)(5,8)(6,7)(9,10)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,31,32,40)(28,34,35,39)(29,37,30,38)(33,41,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012989);