Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #4427 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C32 |
| direct product | C8×C7⋊H(3,𝔽3) |
| unambiguous semidirect product | (C7×C8)⋊H(3,𝔽3) |
| unambiguous semidirect product | (C32×C8)⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C3×C8×ΓL(1,𝔽8))⋊C3 |
| Fitting | C32×C7×C8 ⋊ C3 |
| Hall | C7×C8 ⋊ H(3,𝔽3) |
| Sylow | C7 ⋊ C8×H(3,𝔽3) |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5,6,7,8)(9,10,11,13,12,14,15)(17,20,21)(19,23,24);(10,11,12)(13,15,14)(16,17,19)(18,21,23)(20,24,22)232 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 63);