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×#53293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4782 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)(11,12,16,13)(14,19,21,17)(15,20,23,18)(22,24,26,25)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,42)(39,41);(4,7)(8,10)(12,13)(14,21)(15,23)(24,25)(27,29,32,37)(28,33,35,30)(31,36,39,42)(34,40,41,38);(3,4)(5,8)(6,7)(9,10)(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,29)(28,30)(31,36)(32,37)(33,35)(34,38)(39,42)(40,41);(3,5)(4,8)(6,9)(7,10)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,30)(28,29)(31,38)(32,33)(34,36)(35,37)(39,40)(41,42);(3,4)(5,8)(6,7)(9,10)(11,15,16,23)(12,18,13,20)(14,22,21,26)(17,24,19,25)(27,31,32,39)(28,34,35,41)(29,36,37,42)(30,38,33,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012962);