Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of SL(2,𝔽8) |
| automorphism | Aut(G) = ΣL(2,𝔽8) (order 1512) · Inn(G) = G · Out(G) = C3 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
| projective special linear | PSL(2,𝔽8) |
| special linear | SL(2,𝔽8) |
| projective general linear | PGL(2,𝔽8) |
| symplectic | Sp(2,𝔽8) |
| projective symplectic | PSp(2,𝔽8) |
| special unitary | SU(2,𝔽8) |
| projective special unitary | PSU(2,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,9,8,6,5,4,2);(1,2,8,9,7,5,3)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 56 | 56 | 56 | 56 | 63 | 72 | 72 | 72 |
| order | 1 | 3 | 9 | 9 | 9 | 2 | 7 | 7 | 7 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 7 | 1 | −ζ4+ζ8−ζ13−ζ16+ζ17−ζ22−ζ25+ζ26−ζ31−ζ34−ζ40−ζ43+ζ44−ζ52+ζ53−ζ58−ζ61+ζ62 | ζ4+ζ5+ζ13+ζ22+ζ23+ζ31+ζ32+ζ40+ζ41+ζ50+ζ58+ζ59 | −ζ5−ζ8+ζ16−ζ17−ζ23+ζ25−ζ26−ζ32+ζ34−ζ41+ζ43−ζ44−ζ50+ζ52−ζ53−ζ59+ζ61−ζ62 | -1 | 0 | 0 | 0 |
| χ3 | 7 | 1 | ζ4+ζ5+ζ13+ζ22+ζ23+ζ31+ζ32+ζ40+ζ41+ζ50+ζ58+ζ59 | −ζ5−ζ8+ζ16−ζ17−ζ23+ζ25−ζ26−ζ32+ζ34−ζ41+ζ43−ζ44−ζ50+ζ52−ζ53−ζ59+ζ61−ζ62 | −ζ4+ζ8−ζ13−ζ16+ζ17−ζ22−ζ25+ζ26−ζ31−ζ34−ζ40−ζ43+ζ44−ζ52+ζ53−ζ58−ζ61+ζ62 | -1 | 0 | 0 | 0 |
| χ4 | 7 | 1 | −ζ5−ζ8+ζ16−ζ17−ζ23+ζ25−ζ26−ζ32+ζ34−ζ41+ζ43−ζ44−ζ50+ζ52−ζ53−ζ59+ζ61−ζ62 | −ζ4+ζ8−ζ13−ζ16+ζ17−ζ22−ζ25+ζ26−ζ31−ζ34−ζ40−ζ43+ζ44−ζ52+ζ53−ζ58−ζ61+ζ62 | ζ4+ζ5+ζ13+ζ22+ζ23+ζ31+ζ32+ζ40+ζ41+ζ50+ζ58+ζ59 | -1 | 0 | 0 | 0 |
| χ5 | 7 | -2 | 1 | 1 | 1 | -1 | 0 | 0 | 0 |
| χ6 | 8 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 |
| χ7 | 9 | 0 | 0 | 0 | 0 | 1 | −ζ3−ζ24−ζ39−ζ60 | −ζ12−ζ30−ζ33−ζ51 | −ζ6−ζ15−ζ48−ζ57 |
| χ8 | 9 | 0 | 0 | 0 | 0 | 1 | −ζ12−ζ30−ζ33−ζ51 | −ζ6−ζ15−ζ48−ζ57 | −ζ3−ζ24−ζ39−ζ60 |
| χ9 | 9 | 0 | 0 | 0 | 0 | 1 | −ζ6−ζ15−ζ48−ζ57 | −ζ3−ζ24−ζ39−ζ60 | −ζ12−ζ30−ζ33−ζ51 |
Here ζ is a primitive root of unity of order 63 (ζ63 = 1).
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 156);