Nicola's Jigsaw

Age 11 to 14 Challenge Level:
Five 3D jigsaw pieces made from multilink
Nicola has a three dimensional puzzle made from cubes.
  • There are six pieces altogether
  • All six pieces are different shapes
  • The pieces fit together to make a 3 by 3 by 3 cube.
Here are diagrams showing five of the pieces:

Three small cubes joined in a line Five small cubes, four joined in a square and a fifth sticking out in the same plane (or a row of two on top of a row of three) Eight cubes forming a two by two by two cube Three cubes joined in an L shape A line of two cubes horizontally, joined to the left of the base of a tower of two cubes stacked vertically. 
Unfortunately, the sixth piece has gone missing!

Can you figure out what shape the sixth piece is?

Remember, the six pieces must go together to form a 3 by 3 by 3 cube, and they must all be different. You may wish to use multilink cubes like the ones shown in the photo to help you to solve the problem.

Is there more than one possible shape that the sixth piece could be?

Many thanks to Nicola Coe who created the original task on which this problem is based.