I know that the Loewy series of a group can be computed with the help
of a certain central series (Jennings's thm).
Is there either some feature in GAP which determines this series for a
given group or somebody who already wrote a GAP program for that
purpose?

As for p-groups I have written a GAP procedure to compute the Jennings
series of such groups using the fact that it is isomorphic to the
Lazard series L_n(G) = \prod_{ip^j \ge n}\gamma_i(G)^{p^j}. The
calculations are straightforward in GAP.
Furthermore there is a function to test whether a given pc-presentation
for the group is compatible with the Jennings series. In this case
such a presentation can be used (by a theorem of Jennings) to compute
a basis for the group algebra over GF(p), which is compatible with the
Loewy series. This approach has been implemented in the SISYPHOS
program developped in Stuttgart to investigate characteristic p group
algebras of p-groups. It is planned to distribute the program as a
GAP shared library.

Martin Wursthorn
Mathematisches Institut B
Universit"at Stuttgart