Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#29648 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(10,11,14,17)(12,18,20,15)(13,16,21,23)(19,24,25,22)(26,27,30,34)(28,36,29,32)(31,40,39,35)(33,37,41,38);(6,7)(8,9)(10,12,14,20)(11,15,17,18)(13,19,21,25)(16,22,23,24)(26,28,30,29)(27,32,34,36)(31,37,39,38)(33,35,41,40);(7,9)(10,13)(11,16)(12,19)(14,21)(15,22)(17,23)(18,24)(20,25)(26,29,30,28)(27,33,34,41)(31,37,39,38)(32,35,36,40);(13,21)(16,23)(19,25)(22,24)(28,29)(32,36)(33,41)(37,38)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22636);