We demonstrate a compact magneto-optical trap (MOT) of alkaline-earth atoms using a nanofabricated diffraction grating chip. A single input laser beam, resonant with the broad $^1$S$_0,rightarrow ,^1$P$_1$ transition of strontium, forms the MOT in combination with three diffracted beams from the grating chip and a magnetic field produced by permanent magnets. A differential pumping tube limits the effect of the heated, effusive source on the background pressure in the trapping region. The system has a total volume of around 2.4 L. With our setup, we have trapped up to $5 times 10^6$ $^{88}$Sr atoms, at a temperature of approximately $6$ mK, and with a trap lifetime of approximately 1 s. Our results will aid the effort to miniaturize optical atomic clocks and other quantum technologies based on alkaline-earth atoms.
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