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) = C1 |
| direct product | C5×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | D4 |
| commutator | C4 | C2×C5×C8 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| socle | C2×C5 | C4⋊C8 |
| prime radical | C22×C5 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×C8 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,5,8,6,9,3,11,7,10,12,15,4,14,13,16)(17,18,19,20,21);(1,3,4,5,7,13,6,12)(2,8,9,11,10,15,14,16)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 169);