Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C7⋊C8)⋊C27 |
| unambiguous semidirect product | (C7⋊C27)⋊C8 |
| derived, Sylow | C7 ⋊ C8×C27 |
| Hall | C7⋊C27 ⋊ C8 |
| Hall | C7⋊C8 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4×C9 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C8×C27 |
| Fitting, cosocle | C4×C7×C9 | C2×C3 |
| socle, prime radical | C2×C3×C7 | C4×C9 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C4×C9 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1512 |
Polycyclic presentation:
2,3,2,3,2,3,7|a^2=c,b^3=d,c^2=e,d^3=f,g^a=g^6,g^b=g^2252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 1);