Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10447 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15,19,23)(16,24,27,39)(17,28,31,21)(18,32,34,22)(20,33,36,26)(25,41,29,40)(30,38,45,43)(35,42,37,44);(7,10)(11,12)(14,16,19,27)(15,20,23,36)(17,29,31,25)(18,33,34,26)(21,37,28,35)(22,39,32,24)(30,42,45,44)(38,40,43,41);(7,11)(9,13)(10,12)(14,17,18,30,19,31,34,45)(15,21,22,38,23,28,32,43)(16,25,26,42,27,29,33,44)(20,35,24,40,36,37,39,41)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3435);