Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53302 |
| 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 | |
4570 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)(8,10)(11,12,16,14)(13,19,21,17)(15,22,23,18)(20,24,26,25)(27,28,31,35)(29,33,37,32)(30,38,40,34)(36,41,39,42);(4,7)(8,10)(12,14)(15,23)(17,19)(20,26)(27,29,31,37)(28,32,35,33)(30,39,40,36)(34,42,38,41);(3,4)(5,8)(6,7)(9,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,29)(28,33)(30,36)(31,37)(32,35)(34,42)(38,41)(39,40);(3,5)(4,8)(6,9)(7,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30,31,40)(28,34,35,38)(29,36,37,39)(32,41,33,42);(3,4)(5,8)(6,7)(9,10)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,28)(29,33)(30,34)(31,35)(32,37)(36,42)(38,40)(39,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012888);