mathematics - Three-digit multiplication puzzle, part II: ever heard of senary?

Followup to: Three-digit multiplication puzzle

I'd never heard of "senary" either, until creating this series of three puzzles. It seems that "senary" is the word for base-six notation. Who knew? Without further ado, here's part II:

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Place different three-digit senary numbers (000-555) on each of the seven nodes of the following diagram:

There are six lines in the diagram, on each of which are three of the nodes. On each of those lines, the result of multiplying the numbers on those nodes should end in 001 in senary. The same thing for the circle, on which are also three of the nodes.

Answer

One possible arrangement for the solution not given by the OP is:

and we can note:

$025_6+531_6=1000_6$
$101_6+455_6=1000_6$
$231_6+325_6=1000_6$

leaving $155_6$ unpaired.