Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C22⋊C4×C27 ⋊ C2 |
| Sylow | C27 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D27 |
| commutator, cosocle | C22×C27 | C24 |
| Frattini | C22×C9 | C23×D3 |
| Fitting | C2×C22⋊C4×C27 | C2 |
| socle | C22×C3 | C23×D9 |
| hypercenter | C2×C22⋊C4 | D27 |
| hypocenter | C27 | #538 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
500 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^6,3^3|c^2=e,g^3=h,h^3=i,b^a=bf,d^a=de,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2,d^c=df282 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2295);