Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #464 |
390 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|b^2=f,c^2=f,e^2=f,b^a=bd,c^a=ce,e^a=ef,g^a=g^2,h^a=h^2,i^a=h^2i,c^b=ce,e^b=ef,h^b=h^2i^2,e^c=ef,h^d=h^2,i^d=i^266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 33319);