Small oligomers formed early in the process of amyloid fibril formation may be the major toxic species in Alzheimers disease. We investigate the early stages of amyloid aggregation for the tau fragment AcPHF6 (Ac-VQIVYK-NH2) using an implicit solvent all-atom model and extensive Monte Carlo simulations of 12, 24, and 36 chains. A variety of small metastable aggregates form and dissolve until an aggregate of a critical size and conformation arises. However, the stable oligomers, which are beta-sheet-rich and feature many hydrophobic contacts, are not always growth-ready. The simulations indicate instead that these supercritical oligomers spend a lengthy period in equilibrium in which considerable reorganization takes place accompanied by exchange of chains with the solution. Growth competence of the stable oligomers correlates with the alignment of the strands in the beta-sheets. The larger aggregates seen in our simulations are all composed of two twisted beta-sheets, packed against each other with hydrophobic side chains at the sheet-sheet interface. These beta-sandwiches show similarities with the proposed steric zipper structure for PHF6 fibrils but have a mixed parallel/antiparallel beta-strand organization as opposed to the parallel organization found in experiments on fibrils. Interestingly, we find that the fraction of parallel beta-sheet structure increases with aggregate size. We speculate that the reorganization of the beta-sheets into parallel ones is an important rate-limiting step in the formation of PHF6 fibrils.