Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of AΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | AGL(1,𝔽8)⋊C27 |
| derived, Hall | AGL(1,𝔽8) ⋊ C27 |
| Sylow | C23 ⋊ C7⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C9 | AΓL(1,𝔽8) |
| commutator, hypocenter | AGL(1,𝔽8) | C27 |
| Fitting | C23×C9 | ΓL(1,𝔽8) |
| socle | C23×C3 | C7⋊C9 |
| prime radical | C3×AGL(1,𝔽8) | C9 |
| cosocle | C9×AGL(1,𝔽8) | C3 |
| LCS | G ▷ AGL(1,𝔽8) | |
| UCS | C1 ◁ C9 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
3^3,7,2^3|a^3=b,b^3=c,d^a=d^2,e^a=f,f^a=ef,e^d=eg,f^d=ef,g^d=f72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 82);