On top of my head this should work - of course look through all the commands in our command reference in the help manual.
So first select the element, select the nodes connected to it, and get the number of nodes:
ElNumber = 1 ! element number to select
esel,s,elem,,ElNumber ! select elment
nsle,s ! select nodes connected to it
*GET, MaxNodes, NODE, 0, count, ! get the number if nodes
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