Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/16) (order 256) · Inn(G) = D8 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D8 |
| commutator | C8 | C2×C4 |
| Frattini, cosocle | C2×C8 | C22 |
| socle | C2 | #107 |
| prime radical | C22 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,7,11,6,21)(3,12,15,27,17,28,16,31)(4,18,19,25,20,32,13,9)(8,22,23,30,24,29,26,14);(1,3,13,29,6,16,20,30,7,17,19,22,5,15,4,14)(2,8,18,27,10,23,25,28,11,24,32,31,21,26,9,12)Polycyclic presentation:
2^6|a^2=d,b^2=ce,c^2=ef,d^2=f,e^2=f,b^a=bc,c^a=ce,e^a=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | #1417, #1419, #1420, … |
| maximal normal subgroup of | #1417, #1419, #1420, … |
| maximal quotient of | #1371, #1372, #1381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 49);