first thought would be to verify that the task is assigned to the same user that is used to logon to jbpm console.
You can always give it a try with api access (TaskService) to look up for tasks using REST interface of gwt-server-console.
No the task is not assigned yet. In the database the ASSIGNEE_ column is NULL. What I was hoping for is to see it under the "Group Tasks", select it, and then click on the "Claim" button to assign the task to myself. The problem is, it's not showing up under the "Group Tasks".
Did you assign task to a group (using candidate-group attribute in process definition)? How does your process definition for this task look like?
Since it is not assigned to anything then it will not show up in the console. If you want to have it assigned to a group and then to be shown in console user that you are logging in with must be part of this group.