# SmallGroup(256,6745)

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## Definition

This group of order 256 is defined by means of the following presentation (here $e$ denote the identity element): $G := \langle a_1, a_2, a_3, a_4 \mid a_1^4 = a_2^4 = a_3^4 = a_4^4 = e, [a_1,a_2] = a_3^2, [a_1,a_3] = e, [a_2,a_3] = a_4^2, [a_1,a_4] = [a_2,a_4] = [a_3,a_4] = e \rangle$

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
Function Value Similar groups Explanation for function value
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 256 groups with same order
prime-base logarithm of order 8 groups with same prime-base logarithm of order
max-length of a group 8 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 8 chief length equals prime-base logarithm of order for group of prime power order
composition length 8 composition length equals prime-base logarithm of order for group of prime power order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 2 groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class 2 groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length 2 groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length

## 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 $G$:

gap> G := SmallGroup(256,6745);

Conversely, to check whether a given group $G$ 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 ]