Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D189 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊(C27⋊C8) |
| unambiguous semidirect product | C27⋊(C7⋊C8) |
| derived, Hall | C7×C27 ⋊ C8 |
| Sylow | C27 ⋊ C7⋊C8 |
| Sylow | C7 ⋊ C27⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D189 |
| commutator, hypocenter | C7×C27 | C8 |
| Frattini | C4×C9 | D21 |
| Fitting, cosocle | C4×C7×C27 | C2 |
| socle, prime radical | C2×C3×C7 | Dic9 |
| LCS | G ▷ C7×C27 | |
| UCS | C1 ◁ C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1512 |
Polycyclic presentation:
2^3,3,7,3^2|a^2=b,b^2=c,d^3=f,f^3=g,d^a=d^2f^2g^2,e^a=e^6,f^a=f^2g^2,g^a=g^2384 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 5);