Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33589 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
726 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)(7,9)(10,11)(12,17)(13,19)(14,27)(15,31)(16,21)(18,23)(20,29)(22,32)(24,35)(25,34)(26,38)(28,33)(30,41)(36,40)(37,39);(6,7,8,9)(10,12)(11,17)(13,25)(14,23)(15,22)(16,24)(18,27)(19,34)(20,33)(21,35)(26,40)(28,29)(30,39)(31,32)(36,38)(37,41);(10,12)(11,18)(13,22)(14,28)(15,25)(16,24)(17,27)(19,33)(20,34)(21,37)(23,29)(26,39)(30,40)(31,36)(32,38)(35,41);(10,13,16,15)(11,19,21,31)(12,22,24,25)(14,26,30,29)(17,32,35,34)(18,33,37,36)(20,27,38,41)(23,28,39,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26577);