# SmallGroup(256,6745)

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## Contents

## Definition

This group of order 256 is defined by means of the following presentation (here denote the identity element):

Here denotes the commutator of two elements. Because the group has nilpotency class two, the choice of convention for commutator (left or right) does not matter.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 256#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 256 and has ID 6745 among the groups of order 256 in GAP's SmallGroup library. For context, there are groups of order 256. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(256,6745)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(256,6745);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [256,6745]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Description by presentation

gap> F := FreeGroup(4); <free group on the generators [ f1, f2, f3, f4 ]> gap> G := F/[F.1^4,F.2^4,F.3^4,F.4^4,Comm(F.1,F.2)*F.3^(-2),Comm(F.1,F.3),Comm(F.2,F.3)*F.4^(-2),Comm(F.1,F.4),Comm(F.2,F.4),Comm(F.3,F.4)]; <fp group on the generators [ f1, f2, f3, f4 ]> gap> IdGroup(G); [ 256, 6745 ]