A multinucleon transfer reaction between a thin self-supporting Pt-198(78) target and an 850 MeV Xe-136(54) beam has been used to populate and study the structure of the N=80 isotone Ba-136(56). Making use of time-correlated gamma-ray spectroscopy, evidence for an I-pi=(10(+)) isomeric state has been found with a measured half-life of 91+/-2 ns. Prompt-delayed correlations have also enabled the tentative, measurement of the near-yrast states which lie above the isomer. Shell-model calculations suggest that the isomer has a structure which can be assigned predominantly as (nuh(11/2))(10+)(-2). The results are discussed in terms of standard and pair-truncated shell-model calculations, and compared to the even-Z N=80 isotones ranging from Sn-130(50) to Er-148(68). A qualitative explanation of the observed dramatic decrease in the B(E2:10(+)-->8(+)) value for the N=80 isotones at Ba-136 is given in terms of the increasing single-hole energy of the h(11/2) neutron configuration as the proton subshell is filled. The angular momentum transfer to the binary fragments in the reaction has also been investigated in terms of the average total gamma-ray fold versus the scattering angle of the recoils.