As you expressed interest in Thompson's group V in the comments --
the following is a highly transitive faithful permutation representation of V on $\mathbb{Z}$ (although as Ives de Cornulier has already mentioned this group is *not*
a torsion group):

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the *class transposition*
$\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges
$r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

Then we have $V \cong \langle \kappa, \lambda, \mu, \nu \rangle$, where
$\kappa := \tau_{0(2),1(2)}$, $\lambda := \tau_{1(2),2(4)}$, $\mu := \tau_{0(2),1(4)}$
and $\nu := \tau_{1(4),2(4)}$ satisfy the defining relations

$\kappa^2 = \lambda^2 = \mu^2 = \nu^2 = 1$,

$\lambda\kappa\mu\kappa\lambda\nu\kappa\nu\mu\kappa\lambda\kappa\mu = 1$,

$\kappa\nu\lambda\kappa\mu\nu\kappa\lambda\nu\mu\nu\lambda\nu\mu = 1$,

$(\lambda\kappa\mu\kappa\lambda\nu)^3 = (\mu\kappa\lambda\kappa\mu\nu)^3 = 1$,

$(\lambda\nu\mu)^2\kappa(\mu\nu\lambda)^2\kappa = 1$,

$(\lambda\nu\mu\nu)^5 = 1$,

$(\lambda\kappa\nu\kappa\lambda\nu)^3\kappa\nu\kappa(\mu\kappa\nu\kappa\mu\nu)^3 \kappa\nu\kappa\nu = 1$,

$((\lambda\kappa\mu\nu)^2(\mu\kappa\lambda\nu)^2)^3 = 1$,

$(\lambda\nu\lambda\kappa\mu\kappa\mu\nu\lambda\nu\mu\kappa\mu\kappa)^4 = 1$,

$(\mu\nu\mu\kappa\lambda\kappa\lambda\nu\mu\nu\lambda\kappa\lambda\kappa)^4 = 1$,

$(\lambda\mu\kappa\lambda\kappa\mu\lambda\kappa\nu\kappa)^2 = 1$, and

$(\mu\lambda\kappa\mu\kappa\lambda\mu\kappa\nu\kappa)^2 = 1$

given in:

Graham Higman, *Finitely presented infinite simple groups*, Notes on Pure
Mathematics, Department of Pure Mathematics, Australian National University,
Canberra, 1974. MR0376874.

3more comments