Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15)(16,25)(17,30)(18,35)(19,23)(20,24)(21,27)(22,33)(26,41)(28,37)(29,40)(31,36)(32,39)(34,44)(38,43)(42,45);(7,10)(11,12)(14,16,20,29)(15,21,24,37)(17,31,33,42)(18,28,19,27)(22,38,30,41)(23,40,35,25)(26,32,43,34)(36,44,45,39);(7,11)(9,13)(10,12)(14,17,18,32,20,33,19,34)(15,22,23,39,24,30,35,44)(16,26,27,42,29,43,28,31)(21,36,25,38,37,45,40,41)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3455);