Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C5 |
| prime radical | C23×C5 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,27,28,29);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24);(2,6)(5,8)(11,18)(14,17)(19,24)(22,23);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1543);