  Utilizing the first-principles calculations together with the group theory
analysis, we systematically investigate the magnetocrystalline anisotropy
energy, magneto-optical effect, and anomalous transport properties (including
anomalous Hall, Nernst, and thermal Hall effects) of monolayer and bilayer
Fe$_n$GeTe$_2$ ($n$ = 3, 4, 5). The monolayer Fe$_n$GeTe$_2$ ($n$ = 3, 4, 5)
exhibits the out-of-plane, in-plane, and in-plane ferromagnetic orders with
considerable magnetocrystalline anisotropy energies of -3.17, 4.42, and 0.58
meV/f.u., respectively. Ferromagnetic order is predicted in bilayer
Fe$_4$GeTe$_2$ while antiferromagnetic order prefers in bilayer Fe$_3$GeTe$_2$
and Fe$_5$GeTe$_2$. The group theory analysis reveals that in addition to
monolayer ferromagnetic Fe$_n$GeTe$_2$ ($n$ = 3, 4, 5), the magneto-optical and
anomalous transport phenomena surprisingly exist in bilayer antiferromagnetic
Fe$_5$GeTe$_2$, which is much rare in realistic collinear antiferromagnets. If
spin magnetic moments of monolayer and bilayer Fe$_n$GeTe$_2$ are reoriented
from the in-plane to out-of-plane direction, the magneto-optical and anomalous
transport properties enhance significantly, presenting strong magnetic
anisotropy. We also demonstrate that the anomalous Hall effect decreases with
the temperature increases. The gigantic anomalous Nernst and thermal Hall
effects are found in monolayer and bilayer ferromagnetic Fe$_n$GeTe$_2$, and
the largest anomalous Nernst and thermal Hall conductivities, respectively, of
-3.31 A/Km and 0.22 W/Km at 130 K are observed in bilayer ferromagnetic
Fe$_4$GeTe$_2$. Especially, bilayer antiferromagnetic Fe$_5$GeTe$_2$ exhibits
large zero-temperature anomalous Hall conductivity of 2.63 e$^2$/h as well as
anomalous Nernst and thermal Hall conductivities of 2.76 A/Km and 0.10 W/Km at
130 K, respectively.
