We consider the near-rings generated by endomorphisms of some extra-special2-groups. The most essential difference of a near-ring from a usual ring is the absence of thesecond distributivity. In this paper, we prove that the near-ring E(G) generated by endomorph-isms of an extra-special 2-group G of order 2~(2n+1)has the order which divides2~(22n)+4~2andthat the near-ring E(G) of the extra-special 2-group G of type — of order 22п+1 has the orderdivided by 2~(22n+4n~2-2In this case, for n=1 andn =2 the upper bound is attainable: thenear-ring E(G) of the group D8 has the order 28, and the near-ring E(G) of an extra-special2-group D_8*
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