Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C33 ⋊ #359 |
168 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|a^2=c,c^2=e,e^2=f,b^a=bcdef,g^a=g^2,h^a=h^2i^2,i^a=h^2,c^b=ce,d^b=df,e^b=ef,h^b=h^2i,h^c=h^2i,i^c=hi,h^e=h^2,i^e=i^251 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31165);