Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13369 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
378 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)(6,7,9,11)(8,12,13,10)(14,15,17,22)(16,26,28,20)(18,21,31,37)(19,24,32,25)(23,39,38,33)(27,35,30,40)(29,44,43,36)(34,42,41,45);(6,8,9,13)(7,10,11,12)(14,16,17,28)(15,20,22,26)(18,29,31,43)(19,27,32,30)(21,36,37,44)(23,34,38,41)(24,40,25,35)(33,42,39,45);(15,21)(16,27)(18,33)(19,32)(20,34)(22,37)(23,25)(24,38)(26,41)(28,30)(29,43)(31,39)(35,44)(36,40)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6357);