Using data from the GALAH pilot survey, we determine properties of the Galactic thin and thick discs near the solar neighbourhood. The data cover a small range of Galactocentric radius (7.9 ≲ RGC ≲ 9.5 kpc), but extend up to 4 kpc in height from the Galactic plane, and several kpc in the direction of Galactic anti-rotation (at longitude 260° ≤ ℓ ≤ 280°). This allows us to reliably measure the vertical density and abundance profiles of the chemically and kinematically defined 'thick' and 'thin' discs of the Galaxy. The thin disc (low-α population) exhibits a steep negative vertical metallicity gradient, at d[M/H]/dz = -0.18 ± 0.01 dex kpc -1 , which is broadly consistent with previous studies. In contrast, its vertical α-abundance profile is almost flat, with a gradient of d[α/M]/dz = 0.008 ± 0.002 dex kpc -1 . The steep vertical metallicity gradient of the low-α population is in agreement with models where radial migration has a major role in the evolution of the thin disc. The thick disc (high-α population) has a weaker vertical metallicity gradient d[M/H]/dz = -0.058 ± 0.003 dex kpc -1 . The α-abundance of the thick disc is nearly constant with height, d[α/M]/dz = 0.007 ± 0.002 dex kpc -1 . The negative gradient in metallicity and the small gradient in [α/M] indicate that the high-α population experienced a settling phase, but also formed prior to the onset of major Type Ia supernova enrichment. We explore the implications of the distinct α-enrichments and narrow [α/M] range of the sub-populations in the context of thick disc formation.