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_NGC 4589, a bright E2 merger-remnant galaxy, hosts the peculiar fast andfaint calcium-rich Type Ib supernova (SN) SN 2005cz. The progenitor of Ca-richSNe Ib has been controversial: it could be a) a young massive star with 6-12M$\odot$ in a binary system, or b) an old low-mass star in a binary system thatwas kicked out from the galaxy center. Moreover, previous distance estimatesfor this galaxy have shown a large spread, ranging from 20 Mpc to 60 Mpc. Thus,using archival $Hubble$ $Space$ $Telescope$/ACS $F435W$, $F555W$, and $F814W$images, we search for star clusters in NGC 4589 in order to help resolve theseissues. We find a small population of young star clusters with $25<V\leq27$($-7.1<M_V\leq-5.1$) mag and age $< 1$ Gyr in the central region at $R<0.5'$($<3.8$ kpc), thus supporting the massive-star progenitor scenario for SN2005cz. In addition to young star clusters, we also find a large population ofold globular clusters. In contrast to previous results in the literature, wefind that the color distribution of the globular clusters is clearly bimodal.The turnover (Vega) magnitude in the $V$-band luminosity functions of the blue(metal-poor) globular clusters is determined to be $V_0{(\rmmax)}=24.40\pm0.10$ mag. We derive the total number of globular clusters,$N_{\rm GC} =640\pm50$, and the specific frequency, $S_N =1.7\pm0.2$. Adoptinga calibration for the metal-poor globular clusters, $M_V({\rmmax})=-7.66\pm0.14$ mag, we derive a distance to this galaxy:$(m-M)_0=32.06\pm0.10({\rm ran})\pm0.15({\rm sys})$ ($d=25.8\pm2.2$ Mpc).

Star Clusters in the Elliptical Galaxy NGC 4589 Hosting a Calcium-rich SN Ib (SN 2005CZ)