Maximal class group
The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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- The abelianization of the group, i.e., the quotient of the group by its commutator subgroup has order .
- The upper central series and lower central series coincide, and all the successive quotients (except the top-most quotient) are of order .
- Classification of finite 2-groups of maximal class: There are only three possibilities in general for any order -- the dihedral group, semidihedral group, and generalized quaternion group. For order 8, two of these collapse into one and we get only two possibilities.
- Finite non-abelian 2-group has maximal class iff its abelianization has order four: In particular, all the finite 2-groups of a given order and maximal class are isoclinic groups.
- P-group with derived subgroup of prime-square index not implies maximal class for odd p
- Group of exponent p and order greater than p^p is not embeddable in a maximal class group
- P-group is either absolutely regular or maximal class or has normal subgroup of exponent p and order p^p
|Prime number||Nilpotency class||Number of maximal class groups||List of maximal class groups||List of GAP IDs second part (in order of list)|
|2||2||8||2||dihedral group:D8, quaternion group||3,4|
|2||3||16||3||dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16||7,8,9|
|2||4||32||3||dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32||18,19,20|
|2||5||64||3||dihedral group:D64, semidihedral group:SD64, generalized quaternion group:Q64||52,53,54|
|3||2||27||2||prime-cube order group:U(3,3), M27||3,4|
|3||3||81||4||wreath product of Z3 and Z3, SmallGroup(81,8), SmallGroup(81,9), SmallGroup(81,10)||7,8,9,10|
|odd||2||2||prime-cube order group:U(3,p), semidirect product of cyclic group of prime-square order and cyclic group of prime order||3,4|