I want to move all of the pieces of a split hexagon so there is equal space between each piece.
I could use the stroke function but I need it to have a transparent background.
I've split the hexagon into 24 pieces but I am stuck.
Vincent has a great idea, here's another similar method from a different approach (which I think has less steps)
Create your shapes (all split up etc.)
Select All (Ctrl/Command+A)
Then add in an stroke outline and bump it up to the desired thickness (I would make it white to better see it).
Now go to Object→Path→Outline Stroke
Then select your group and open up Pathfinder:
Now select Divide
Now select and ungroup everything (Right-click→Ungroup or Ctrl/Cmd+Shift+G)
Now select one of the white spots and then go to Select→Same→Fill Color and delete all of them.
You will now have equally spaced gaps between the shapes.
The distance between the shapes will depend on your stroke thickness.
Attributes palette (Windows > Attributes) and click the
Show Center button--a square with a dot in the middle. This will cause your hexagon's center to be shown as an anchor-like dot.
line tool (\) and start drawing on the center dot. Displaying your Smart Guides (View > Smart Guides or Ctrl/Cmd+U) helps aligning the start of your line exactly with the center;
Rotate tool (R) and LeftAlt+click on the bottom anchor of the line;
Pathfinder palette (Window > Pathfinder) and choose the first icon, 'unite';
Pathfinder palette, 'Minus Front'.
You can also quickly use the Polar Grid Tool.
Let me know if you need more detailed tutorial or screencast
The easy method which subtract pieces to make the space between the triangles, is already well presented in more than one variations. But if it happens, that you want literally to move the slices as you wrote in the question, you can add a quide shape which has anchor points at the new positions of the triangle heads. You need a 24 side polygon which is distorted to the same height and width proportions than your shape.
In the following cartoon the sliced hexagon is regular, so the 24-gon is regular, too. It's aligned by dragging its centerpoint to the center of the sliced shape.
The slices were selected with the direct selection tool by clicking the fill far away from edges and corners. Then the head anchor point was dragged from the midpoint to the corner of the 24-gon. Snapping to point and smart quides made the exact alignment easy.
The result is totally different than what's got by subtracting the outlined strokes. The spaces between the lines are equal when different spaces are compared. A single space is not uniformly wide.