  Generalized Uncertainty Principle (GUP) was obtained in string theory and
quantum gravity and suggested the existence of a fundamental minimal length
which, as was established, can be obtained within the deformed Heisenberg
algebra. We use the deformed commutation relations or in classical case
(studied in this paper) the deformed Poisson brackets, which are invariant with
respect to the translation in configurational space. We have found
transformations relating coordinates and times of moving and rest frames of
reference in the space with GUP in the first order over parameter of
deformation. For the non-relativistic case we find the deformed Galilean
transformation which is similar to the Lorentz one written for Euclidean space
with signature $(+,+,+,+)$. The role of the speed of light here plays some
velocity $u$ related to the parameter of deformation, which as we estimate is
many order of magnitude larger than the speed of light $u\simeq 1.2 \times
10^{22} c$. The coordinates of the rest and moving frames of reference for
relativistic particle in the space with GUP satisfy the Lorentz transformation
with some effective speed of light. We estimate that the relative deviation of
this effective speed of light $\tilde c$ from $c$ is ${(\tilde c-c)/ c}\simeq
3.5\times 10^{-45}$. The influence of GUP on the motion of particle and the
Lorentz transformation in the first order over parameter of deformation is
hidden in $1/c^2$ relativistic effects.
