Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C27 ⋊ #354 |
140 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=c,d^2=f,g^3=h,h^3=i,b^a=bd,c^a=ce,d^a=df,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2,d^b=def,e^b=ef,d^c=df156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31);