Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C5×#33334 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
1172 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)(8,9)(10,11,13,17,14,18,23,29,15,19,24,30,25,31,35,39)(12,16,21,27,22,28,34,38,20,26,32,36,33,37,40,41);(7,9);(10,12)(11,16)(13,21)(14,22)(15,20)(17,27)(18,28)(19,26)(23,34)(24,32)(25,33)(29,38)(30,36)(31,37)(35,40)(39,41);(12,20)(16,26)(21,32)(22,33)(27,36)(28,37)(34,40)(38,41)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26322);