Final answer:
The secret organization with membership limited to twelve associated with Franklin is not named in the provided references. Franklin was known as a Freemason, which could be what the question refers to, but without additional information, the specific organization cannot be identified.
Step-by-step explanation:
The name of Benjamin Franklin's secret organization, which had a membership limited to twelve, is not explicitly mentioned in the provided references. However, Franklin was known for his involvement with the Freemasons. The Freemasons were a secret fraternal society that promoted Enlightenment principles. It is possible that the society in question might refer to a group within or related to the broader Masonic lodges, but without a specific name given, we cannot definitively identify the organization referred to in this question.
Additionally, Benjamin Franklin is known for his contributions to the founding of the United States, including developing a list of virtues and being inspired by the unity of the Iroquois Nation. However, if the question refers to a different, less well-known society, further details would be needed to provide an accurate answer.