Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,11,15,19)(9,16,17,12)(10,13)(14,18)(20,21)(22,23)(24,33)(25,37,40,47)(26,41)(27,31)(28,32)(29,35)(30,39,46,38)(34,49,51,43)(36,44)(42,45)(48,50,53,52);(6,8)(7,11,13,19)(9,14)(10,15)(12,18,16,21)(17,20)(22,24,26,35,28,36,27,42)(23,29,31,33,32,45,41,44)(25,34,39,52,40,51,38,50)(30,43,47,48,46,49,37,53);(6,9)(7,12)(8,14)(10,17)(11,18)(13,16)(15,20)(19,21)(22,25)(23,30)(24,34)(26,39)(27,38)(28,40)(29,43)(31,47)(32,46)(33,48)(35,52)(36,51)(37,41)(42,50)(44,53)(45,49)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2319);