Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#33509 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C23×C8 |
| prime radical | C2×#121 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(11,12,14,19,15,20,24,32,16,18,25,30,26,31,36,40)(13,17,22,28,23,29,35,39,21,27,33,37,34,38,41,42);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,22)(15,23)(16,21)(18,27)(19,28)(20,29)(24,35)(25,33)(26,34)(30,37)(31,38)(32,39)(36,41)(40,42);(4,7)(8,10)(12,18)(13,21)(19,30)(20,31)(22,33)(23,34)(32,40)(35,41);(3,5,6,9)(4,8,7,10)(13,21)(17,27)(22,33)(23,34)(28,37)(29,38)(35,41)(39,42)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529911);