A finite quandle is simply connected exactly when it is principal and all prime-size abelian cocycles are trivial, which yields classifications for sizes p^2 and p^3.
Die Klassifikation der Gruppen bis zur Ordnung p^5
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abstract
This paper presents a classification of all p-groups of order p^5 up to isomorphism. It contains a full list of their polycyclic presentations, a short introduction to the basic ideas of the methodes used to classify the groups, and a detailed proof of the adequacy and completeness of the classification and presentations presented in the paper.
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On Simply Connected Quandles
A finite quandle is simply connected exactly when it is principal and all prime-size abelian cocycles are trivial, which yields classifications for sizes p^2 and p^3.