Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12613 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C16 | C22⋊C4 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | C8×C22⋊C4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C16 ◁ C22×C5×C16 ◁ G | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7,10,15,11,17,26,37,12,16,27,36,28,38,50,59)(8,18,21,40,22,41,46,62,20,13,29,31,45,32,52,55)(9,14,23,33,24,35,47,57,25,34,48,56,49,58,65,68)(19,39,43,60,44,61,64,69,42,30,51,53,63,54,66,67);(6,8)(7,13,16,18)(9,19)(10,21)(11,22)(12,20)(14,30,34,39)(15,31,36,40)(17,32,38,41)(23,43)(24,44)(25,42)(26,46)(27,29)(28,45)(33,53,56,60)(35,54,58,61)(37,55,59,62)(47,64)(48,51)(49,63)(50,52)(57,67,68,69)(65,66);(6,9,10,23,11,24,26,47,12,25,27,48,28,49,50,65)(7,14,15,33,17,35,37,57,16,34,36,56,38,58,59,68)(8,19,21,43,22,44,46,64,20,42,29,51,45,63,52,66)(13,30,31,53,32,54,55,67,18,39,40,60,41,61,62,69)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5601);