This page of our Gotham Knights guide explains how to complete The Powers Club quest. You'll learn how to solve the puzzles and access to hidden tunnels.
Your destination is near Belfry. After getting inside and eliminated the guards, scan the bloodstains on the floor and the hidden mechanism. Follow the energy lines (white lines) and find the mechanism's activators.
The first mechanism is hidden in the statue on the first floor, while the second one is on the opposite side of the building, on the shelf near the bar. Return to the first room and find a switch behind the wall lamp.
The owl shadow puzzle
After entering the basement you'll have to solve yet another puzzle. Activate the spotlight and set the shadows to match the one pictured above. That's all - once you set them up correctly, the hidden door will open. Defeat a group of enemies on the lower floor - watch out, they can use powerful elemental attacks.
After a cutscene you'll need to get through the trap. There's no pre-determined single safe path - watch out for how the trap's parts move and get through once it's safe, in a slight zig-zag pattern. Once you get through you'll fall down a pit - keep moving forward and jump onto the nearly invisible rock ledge. Finally, return to Belfry.