Measuring rest-fram ultraviolet rotational transitions from the Lyman and Werner bands in absorption against a bright background continuum is one of the few ways to directly measure molecular hydrogen (H2). Here, we report the detection of absorption from H2 at z = 0.56 in a subdamped Lyα system with neutral hydrogen column density NHI = 1019.5±0.2 cm-2. This is the first H2 system analysed at a redshift of <1.5 beyond the Milky Way halo. It has a surprisingly high molecular fraction: log10 fH2 >-1.93 ± 0.36 based on modelling the line profiles, with a robust model-independent lower limit of fH2 > 10-3. This is higher than fH2 values seen along sightlines with similar NHI through the Milky Way disc and the Magellanic Clouds. The metallicity of the absorber is 0.19+0.21-0.10 solar, with a dust-to-gas ratio of <0.36 of the value in the solar neighbourhood. Absorption from associated low-ionization metal transitions such as OI and Fe II is observed in addition to OVI. Using CLOUDY models, we show that there are three phases present; a ̃100 K phase giving rise to H2, a ̃104 K phase where most of the low-ionization metal absorption is produced; and a hotter phase associated with OVI. Based on similarities to high-velocity clouds in the MilkyWay halo showing H2, and the presence of two nearby galaxy candidates with impact parameters of ̃10 kpc, we suggest that the absorber may be produced by a tidally stripped structure similar to the Magellanic Stream.