Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#564 |
| 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 | |
172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(3,6)(4,11)(5,8)(7,13)(9,16)(10,15)(12,14)(17,18,21,24)(19,25,26,22)(20,28,30,23)(27,31,29,32)(33,34,35,36,37);(1,3,5,10)(2,6,8,15)(4,12,13,9)(7,16,11,14)(17,19)(18,22)(20,29)(21,26)(23,32)(24,25)(27,30)(28,31);(1,4,5,13)(2,7,8,11)(3,9,10,12)(6,14,15,16)(19,26)(22,25)(27,29)(31,32);(2,8)(6,15)(7,11)(14,16)(17,20,21,30)(18,23,24,28)(19,27,26,29)(22,31,25,32)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1566);