Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C8×C7⋊C9 |
| unambiguous semidirect product | (C3×C7×C8)⋊C9 |
| derived, Sylow | C7 ⋊ C3×C8×C9 |
| Hall | C7×C8 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C8 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C3×C8×C9 |
| Frattini | C3×C4 | C2×C3×ΓL(1,𝔽8) |
| Fitting | C32×C7×C8 | C3 |
| socle, prime radical | C2×C32×C7 | C3×C4 |
| cosocle | C3×C4×C7 | C2×C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C32×C8 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1512 |
360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 61);