Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×Q8⋊C8 |
| unambiguous semidirect product | Q8⋊(C5×C8) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | D4 |
| commutator | C4 | C2×C5×C8 |
| Frattini, cosocle | C42 | C22×C5 |
| socle, prime radical | C22×C5 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C42×C5 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,5,4,6,7,8)(10,13)(11,15)(14,16)(17,18,19,20,21);(9,10,12,14)(11,16,15,13)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 131);