Unitriangular matrix group:UT(3,8)

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined as the unitriangular matrix group of degree three over field:F8, the unique (up to isomorphism) field with eight elements.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 512#Arithmetic functions

Function	Value	Similar groups	Explanation
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	512	groups with same order	As $UT(n,q),q=8,n=3$ : $q^{n(n-1)/2}=8^{3(2)/2}=512$
prime-base logarithm of order	9	groups with same prime-base logarithm of order
max-length of a group	9		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	9		chief length equals prime-base logarithm of order for group of prime power order
composition length	9		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
Frattini length	2	groups with same order and Frattini length \| groups with same prime-base logarithm of order and Frattini length \| groups with same Frattini length
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	As $UT(n,q),n=3$ : nilpotency class is $n-1=3-1=2$
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	As $UT(n,q),n=3$ : derived length is the smallest integer greater than or equal to $\log _{2}n$ .
minimum size of generating set	6	groups with same order and minimum size of generating set \| groups with same prime-base logarithm of order and minimum size of generating set \| groups with same minimum size of generating set	As $UT(n,q),q=p^{r},n=3,r=3$ : $r(n-1)=3(3-1)=3(2)=6$