Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11007 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1270 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)(10,11)(12,21)(13,26,29,35)(14,30)(15,19)(16,20)(17,23)(18,28,33,27)(22,31,39,41)(24,32)(25,34)(36,38,40,37);(6,7)(8,9)(10,12,16,25)(11,17,20,32)(13,22,29,39)(14,24,15,23)(18,31,33,41)(19,34,30,21)(26,36,35,40)(27,38,28,37);(7,9)(10,13)(11,18)(12,22)(14,28)(15,27)(16,29)(17,31)(19,35)(20,33)(21,36)(23,38)(24,37)(25,39)(26,30)(32,41)(34,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3995);