Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #456 |
550 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|b^2=e,c^2=f,d^2=f,e^2=f,b^a=bd,d^a=df,h^a=h^2,i^a=h^2i,c^b=ce,d^b=df,h^b=h^2i^2,e^c=ef,g^c=g^2,h^d=h^2,i^d=i^263 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31315);