Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = #6342371 (order 512) · Inn(G) = C24 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| prime radical | #120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,8)(3,9,10,6)(5,7,12,14)(11,15,16,13);(1,3,5,11)(2,6,7,13)(4,10,12,16)(8,9,14,15);(2,7)(6,13)(8,14)(9,15);(3,10)(6,9)(11,16)(13,15)Polycyclic presentation:
2^6|a^2=e,b^2=f,b^a=be,c^a=cf,d^b=de25 conjugacy classes — view the full character table.
| maximal subgroup of | #2197, C2×#551, #3475, … |
| maximal normal subgroup of | #2197, C2×#551, #3475, … |
| maximal quotient of | #2342, #2343, #2353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 232);