The answer depends a lot on what kind of description of the radical you ask for. This family of groups of Lie type has been well-studied from the viewpoint of modular representation theory in the defining characteristic (with reference also to the ambient algebraic groups). Even the somewhat degenerate case $p=2$ fits well enough into the general pattern for odd primes.
It's easy to work out explicitly the $p$ irreducible modular representations, for instance using the characteristic 0 model of spaces of homogeneous polynomials in two variables having degree $<p$. Work of Brauer and others filled in the structure of their projective covers in the group algebra; these have very few composition factors. So in this special case you can write down as explicitly as you want all the dimensions involved, including the dimension of the radical (and eventually even its Loewy series). Here are some of the fairly straightforward references, though the story gets far more complicated for groups of higher rank and even for larger finite fields than the prime field:
J.E. Humphreys, Representations of $SL(2, p)$. Amer. Math. Monthly 82 (1975), 21–39.
J.E. Humphreys, Projective modules for $SL(2, q)$. J. Algebra 25 (1973), 513–518.
Henning Haahr Andersen; Jens Jørgensen; Peter Landrock, The projective indecomposable modules of $SL(2, p^n)$. Proc. London Math. Soc. (3) 46 (1983), no. 1, 38–52.