Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C22×C5×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | D4 |
| commutator, Frattini, cosocle | C4 | C24×C5 |
| prime radical | C22×C5×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C4×C5 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(13,14,15,16,17);(3,4);(5,6,7,8,9,10,11,12);(6,8)(7,11)(10,12)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1572);