Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C81 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Q8⋊C81 |
| derived, Sylow | Q8 ⋊ C81 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C27 | A4 |
| commutator, hypocenter | Q8 | C81 |
| Fitting, cosocle | Q8×C27 | C3 |
| socle, prime radical | C2×C3 | C22⋊C27 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C27 | |
19 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
3^4,2^3|a^3=b,b^3=c,c^3=d,e^2=g,f^2=g,e^a=f,f^a=efg,f^e=fg189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 3);