Groups of order 1024: Difference between revisions

Latest revision as of 13:40, 10 November 2023

This article gives information about, and links to more details on, groups of order 1024
See pages on algebraic structures of order 1024 | See pages on groups of a particular order

Statistics at a glance

To understand these in a broader context, see
groups of order 2^n|groups of prime-tenth order

Since $1024=2^{10}$ is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Quantity	Value	Greatest integer function of logarithm of value to base 2	Explanation
Number of groups up to isomorphism	49487367289	35
Number of abelian groups up to isomorphism	42	5	Equals the number of unordered integer partitions of $10$ . See also classification of finite abelian groups.
Number of maximal class groups, i.e., groups of nilpotency class $10-1=9$	3	1	The dihedral group, semidihedral group, and generalized quaternion group; see classification of finite 2-groups of maximal class

GAP implementation

Unfortunately, GAP's SmallGroup library is not available for this order.

@@ Line 2: / Line 2: @@
 ==Statistics at a glance==
+{{quotation|To understand these in a broader context, see<br>[[groups of order 2^n]]<nowiki>|</nowiki>[[groups of prime-tenth order]]}}
 Since <math>1024 = 2^{10}</math> is a [[prime power]], and [[prime power order implies nilpotent]], all groups of this order are [[nilpotent group]]s.
 {| class="sortable" border="1"
-! Quantity !! Value !! Explanation
+! Quantity !! Value !! Greatest integer function of logarithm of value to base 2 !! Explanation
 |-
-| Number of groups up to isomorphism || 49487365422 ||
+| Number of groups up to isomorphism || 49487367289 || 35 ||
 |-
-| Number of [[abelian group]]s up to isomorphism || 42 || Equals the number of [[unordered integer partitions]] of <math>10</math>. See also [[classification of finite abelian groups]].
+| Number of [[abelian group]]s up to isomorphism || 42 || 5 || Equals the number of [[unordered integer partitions]] of <math>10</math>. See also [[classification of finite abelian groups]].
 |-
-| Number of [[maximal class group]]s, i.e., groups of nilpotency class <math>10 - 1 = 9</math> || 3 || The dihedral group, semidihedral group, and generalized quaternion group
+| Number of [[maximal class group]]s, i.e., groups of nilpotency class <math>10 - 1 = 9</math> || 3 || 1 || The dihedral group, semidihedral group, and generalized quaternion group; see [[classification of finite 2-groups of maximal class]]
 |}