Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#359 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D8 |
| commutator | C8 | C2×C4×C5 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| socle | C2×C5 | #102 |
| prime radical | C5×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,5,10,7,11,6,21)(3,12,14,28,15,29,22,26)(4,16,18,27,20,31,19,9)(8,17,23,30,25,32,24,13)(33,34,35,36,37);(1,3)(2,8)(4,17)(5,14)(6,22)(7,15)(9,26)(10,23)(11,25)(12,16)(13,19)(18,30)(20,32)(21,24)(27,28)(29,31)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 164);