Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12843 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,19,16,13,27,20)(12,21,25,37,26,39,41,38)(14,29,30,18,31,33,22,28)(17,32,34,36,23,40,35,24);(10,12,22,40,15,25,14,24,16,26,30,32,27,41,31,36)(11,17,33,37,20,35,18,21,13,23,29,38,19,34,28,39);(7,9)(10,13)(11,16)(12,23)(14,28)(15,19)(17,26)(18,31)(20,27)(21,36)(22,29)(24,39)(25,34)(30,33)(32,37)(35,41)(38,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5831);