To move device from one group to another, you need to go to group management window. Click on left side 'All groups' deck icon. This will open up left side menu showing all groups. On the top, there is a edit icon, click on that. It shows all groups and devices in the system. Select the device you want to move, and click on teh 'Move' button. From the next form select teh group, you want to move the device and click Save button.
The one you attempted earlier from device management page is to onboard a device before it comes to central. So if you assign group to a device from device management page, and the device come up in central with factory default settings, that device automatically go to that group.