Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12939 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C4×D8 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,16,22,17,24,35,23)(12,25,29,41,31,19,30,40)(13,32,27,18,33,38,28,39)(14,34,26,20,15,36,37,21);(7,9)(10,12,16,29,17,31,35,30)(11,18,22,38,24,39,23,32)(13,26,27,15,33,37,28,14)(19,21,40,34,25,20,41,36);(10,13)(11,19)(12,26)(14,30)(15,29)(16,27)(17,33)(18,21)(20,39)(22,40)(23,41)(24,25)(28,35)(31,37)(32,36)(34,38)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5927);