Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #508 |
332 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|b^2=f,c^2=f,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,h^a=h^2,i^a=h^2i,d^b=de,e^b=ef,h^b=h^2i^2,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, 31418);