Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #460 |
1200 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 192 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|a^2=f,b^2=e,c^2=d,e^2=f,b^a=be,e^a=ef,g^a=g^2,h^a=h^2,c^b=cd,i^b=i^2183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 25059);