Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
380 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)(8,15,16,19)(9,12,17,18)(10,13)(11,14)(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);(6,8,10,18,11,16,20,12)(7,9,13,19,14,17,21,15)(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)(8,13)(10,19)(11,17)(14,18)(15,20)(16,21)(22,25,28,40)(23,30,32,46)(24,34,36,51)(26,39,27,38)(29,43,45,49)(31,47,41,37)(33,48,44,53)(35,52,42,50)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2404);