Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #454 |
584 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|b^2=e,c^2=f,e^2=f,h^3=i,b^a=be,c^a=cd,e^a=ef,g^a=g^2,c^b=cd,d^b=df,h^b=h^2i^2,i^b=i^2,e^c=ef123 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 6865);