Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×Q82 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(6,7,8,9)(10,13,12,11);(6,10,8,12)(7,11,9,13);(14,15,16,17)(18,21,20,19);(14,18,16,20)(15,19,17,21)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1560);