  The background of this paper is the following: search of the minimal systems
of generators for this class of group which still was not founded also problem
of representation for this class of group, exploration of systems of generators
for Sylow 2-subgroups $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ of alternating group,
finding structure these subgroups.
  The aim of this paper is to research the structure of Sylow 2-subgroups and
to construct a minimal generating system for such subgroups. In other words,
the problem is not simply in the proof of existence of a generating set with
elements for Sylow 2-subgroup of alternating group of degree $2^k $ and its
constructive proof and proof its minimality. For the construction of minimal
generating set we used the representation of elements of group by automorphisms
of portraits for binary tree.
  The main result is the proof of minimality of this generating set of the
above described subgroups and also the description of their structure. Key
words: minimal system of generators, wreath product, Sylow subgroups, group,
semidirect product.
