Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C5×C27 ⋊ C2 |
| Hall | C5×C27 ⋊ D4 |
| Sylow | C27 ⋊ C22⋊D5 |
| Sylow | C5 ⋊ #6171 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D135 |
| commutator, cosocle | C2×C5×C27 | C22 |
| Frattini | C2×C9 | C2×D15 |
| Fitting | C22×C5×C27 | C2 |
| socle | C2×C3×C5 | C2×D9 |
| hypercenter | C22 | D135 |
| hypocenter | C5×C27 | D4 |
| LCS | G ▷ C2×C5×C27 ▷ C5×C27 | |
| UCS | C1 ◁ C2 ◁ C22 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 540 |
Polycyclic presentation:
2^3,3,5,3^2|d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^4,f^a=f^2g^2,g^a=g^2273 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 29);