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We present the first results of the \textsc{Dragon-II} simulations, a suiteof 19 $N$-body simulations of star clusters with up to $10^6$ stars, with up to$33\%$ of them initially paired in binaries. In this work, we describe the mainevolution of the clusters and their compact objects (COs). All\textsc{Dragon-II} clusters form in their centre a black hole (BH) subsystemwith a density $10-100$ times larger than the stellar density, with the clustercore containing $50-80\%$ of the whole BH population. In all models, the BHaverage mass steeply decreases as a consequence of BH burning, reaching values$\langle m_{\rm BH}\rangle < 15$ M$_\odot$ within $10-30$ relaxation times.Generally, our clusters retain only BHs lighter than $30$ M$_\odot$ over $30$relaxation times. Looser clusters retain a higher binary fraction, because insuch environments binaries are less likely disrupted by dynamical encounters.We find that BH-main sequence star binaries have properties similar to recentlyobserved systems. Double CO binaries (DCOBs) ejected from the cluster exhibitlarger mass ratios and heavier primary masses than ejected binaries hosting asingle CO (SCOBs). Ejected SCOBs have BH masses $m_{\rm BH} = 3-20$ M$_\odot$,definitely lower than those in DCOBs ($m_{\rm BH} = 10-100$ M$_\odot$).

The Dragon-II simulations -- I. Evolution of single and binary compact objects in star clusters with up to 1 million stars