Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#550 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
312 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)(3,4)(6,10)(11,12)(13,14,15,16,17);(5,6,9,10)(7,11,8,12);(2,4)(5,7,9,8)(6,11,10,12);(5,8,9,7)(6,11,10,12)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1552);