Dear Mrs. and Mr. Forum,
Thomas Breuer already answered the question of Thierry Dana-Picard.
however, there are two minor comments, I'd like to add:

When the ``easy'' invariants (orders, centralizers, permutation
character etc.) are not sufficient, and one has to compute the character
table of <g>, it may be sufficient, to compute this table only
partially (especially, if the group is large, the computation of the
whole table is not feasible at all).

A lot of this identification process could be automized. The routine
would be given a group, a character table and (if possible) further
informations, as the type of the permutation character (when using
permutation representations of small degree, this character is sometimes
determined unique by the degree, even without knowing the values on the
conjugacy classes). The routine then would try to find a unique mapping
and compute further information (conjugacy classes, powermap, partial
character table) etc. itself until the correspondence is determined
completely (As the character table is known already, some of these
computations would be much more easy, then in the general case, as
sometimes the character table will give information, that can save
tedious computations, e.g. conjugacy tests).

Some time ago, I had considered writing a routine of this type, since I
needed something similar, but for the particular case, working ``by
hand'' had been faster. If there is sufficiently demand for that kind of
routine, I would reconsider my decision. (I can not promise anything
about availiability of a routine of this type yet.)
In case, you would be interested in such a routine please send me a
short eMail message (to the adress in the footer, not to the gap-forum
itself, as most readers won't be interested in particular answers).

Alexander Hulpke
(hulpke@bert.math.rwth-aachen.de)