Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = AGL(1,ℤ/32) (order 512) · Inn(G) = D16 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic16 |
| quaternion | Q64 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2 | D16 |
| commutator, Frattini, cosocle | C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
15 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)Polycyclic presentation:
2^6|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,c^b=cd,d^b=de,e^b=ef19 conjugacy classes — view the full character table.
| maximal subgroup of | SD128, Q128, C2×Q64, … |
| maximal normal subgroup of | SD128, Q128, C2×Q64, … |
| maximal quotient of | #1416, #1423, C2×Q64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 54);