Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×#41×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C9 | C25×C3 |
| Frattini, socle | C2×C3 | C24×C3×D3 |
| Fitting | C2×#41×3−1+2 | C2 |
| hypercenter | C2×#41 | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×#585 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
2512 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3,2,3^2|c^2=g,h^3=i,e^a=eg,h^a=h^2i^2,i^a=i^2,d^b=dg,h^f=hi^2182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31009);