Use some flinging and cheeky portal usage to get up near the exit. Then jump across to where a hand button is. Press it and crawl through the hole it makes.
Now you're trapped in this room with turrets, trying to get that cube. Use portals to guide the pellets to knock the turrets off. After that, send a pellet on its way to the receiver behind the door. Once it gets close, portal up to the button and press it to open the door for it.
When you miss jumping for the cube, you fall into a place with a button. Press the button, portal back up to the main are and then fling from the new slanted white wall. Be quick though, it goes back after a few seconds. After that, grab the cube, press the button and escape this room. Put the cube on the button and leave.
Time for some mid air flinging. Use the ditch to get yourself started. While mid air, change your portals so you come out of the next platform, keep doing that until you reach the top.