Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#20399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,21)(13,18)(14,19)(15,29)(16,20)(17,24)(22,35)(23,36)(25,31)(26,32)(27,34)(28,33)(30,38)(37,41)(39,40);(10,12,15,25,16,24,14,23)(11,17,19,31,20,21,29,36)(13,22,26,37,27,38,28,39)(18,30,33,41,34,35,32,40);(6,8,9,7)(10,13,16,27)(11,18,20,34)(12,22,24,38)(14,28,15,26)(17,30,21,35)(19,33,29,32)(23,39,25,37)(31,41,36,40);(10,14,16,15)(11,19,20,29)(12,23,24,25)(13,26,27,28)(17,31,21,36)(18,32,34,33)(22,37,38,39)(30,40,35,41)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13387);