  Aims: Investigate properties of a cluster of intermediate-mass black holes
surrounding a supermassive black hole. Methods: We simulate clusters of
equal-mass intermediate-mass black holes ($m_{\rm{IMBH}} = 10^{3}$
${\rm{M_\odot}}$) initialised in a shell between $0.15\leq r$ [pc] $\leq 0.25$
centered about a supermassive black hole. We explore the influence of the
cluster population and supermassive black hole mass on the merger rate, the
ejection rate and the escape velocity. For $M_{\text{SMBH}} = 4\times10^{6}$
${\rm {M}_\odot}$, we use both a Newtonian and post-Newtonian formalism, going
up to the 2.5th order and including cross-terms. For the other two SMBH masses
($M_{\rm{SMBH}} = 4\times10^{5}$ ${\rm{M_\odot}}$ and $M_{\rm{SMBH}} =
4\times10^{7}$ $\rm{M_\odot}$), we model the system only taking into account
relativistic effects. The simulations end once a black hole escapes the
cluster, a merger occurs, or the system has evolved till $100$ Myr. Results:
The post-Newtonian formalism accelerates the loss rate of intermediate-mass
black holes. Ejections occur more often for lower supermassive black hole
masses while more massive ones increase the rate of mergers. Although
relativistic effects allow for circularisation, all merging binaries have $e
\gtrsim 0.97$. Strong gravitational wave signals are suppressed during our
Newtonian calculations. Weaker and more frequent signals are expected from
gravitational wave radiation emitted in a fly-by. In our post-Newtonian
calculations, $30/406$ of the gravitational wave events capable of being
observed with LISA and $\mu$Ares are detected as gravitational wave capture
binaries with the remaining being in-cluster mergers. Throughout our
investigation, no IMBH-IMBH binaries were detected.
