Thanks for the remark about runtimes: it made me think: going through all permutations of all cubes would be horrendous.  Much better to list all the cubes of a given number of digits and check pairs for being anagrams.

The first found is the 12-digit 127035954683, which has four other permutations that are cubes: 352045367981 = 7061^3, 373559126408 = 7202^3, 569310543872 = 8288^3 and 589323567104 = 8384^3.

The next set of 5 is 140283769536, 536178930624, 613258407936, 913237656408 and 936302451687.

If we want a cube with five other permutations that are cubes, it would be the 13-digit 1000600120008, which has permutations 1006012008000, 1061208000000, 8001200060001, 8012006001000 and 8120601000000.  Next would be the set 1426487591593, 1432197595648, 3496581419752, 4275981654391, 4813967954125 and 7591941538264.

The above were all found within a runtime of less than one minute. It took only a minute or two more to check the rest of the 13-digit cubes, which included only three other sets of five:

1961574655832 2981631556457 5657831164259 8631525145697 9625537115648

2097643558401 2501609484375 2595641083407 6140550947328 9083540714625

2546097986375 2567976059384 5429503678976 6587579024936 7736546098952

DefDbl A-Z

Dim crlf$, cube(20000) As String, hit(10) As String

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 For digs = 2 To 13

   DoEvents

   lower = -Int(-10 ^ ((digs - 1) / 3))

   upper = Int((10 ^ digs - 1) ^ (1 / 3))

   Text1.Text = Text1.Text & digs & Str(lower) & Str(upper) & Str(upper - lower + 1) & crlf

   

   cubect = 0

   For cr = lower To upper

     cubect = cubect + 1

     cube(cubect) = LTrim(Str(cr * cr * cr))

   Next

   

   For i = 1 To cubect

     DoEvents

     hitct = 0

     For j = i + 1 To cubect

       If isPerm(cube(i), cube(j)) Then hitct = hitct + 1: hit(hitct) = cube(j)

     Next

     If hitct >= 4 Then

       Text1.Text = Text1.Text & cube(i)

       For j = 1 To hitct

         Text1.Text = Text1.Text & " " & hit(j)

       Next

       Text1.Text = Text1.Text & crlf

     End If

   Next i

   Text1.Text = Text1.Text & crlf

 Next

 

 

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub

Function isPerm(a$, b$)

  x$ = a$

  good = 1

  If Len(a$) <> Len(b$) Then isPerm = 0: Exit Function

  For i = 1 To Len(b$)

    ix = InStr(x, Mid(b, i, 1))

    If ix = 0 Then good = 0: Exit For

    x = Left(x, ix - 1) + Mid(x, ix + 1)

  Next

  isPerm = good

End Function