Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3280 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
686 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)(10,11,15,20)(12,21,25,32)(13,26,29,35)(14,17,28,19)(16,31,34,23)(18,36,38,27)(22,39,40,33)(24,41,30,37);(10,12,15,25)(11,16,20,34)(13,22,29,40)(14,30,28,24)(17,33,19,39)(18,37,38,41)(21,35,32,26)(23,36,31,27);(7,9);(11,17)(12,22)(13,27)(14,28)(16,32)(19,20)(21,34)(23,24)(25,40)(26,35)(29,36)(30,31)(33,37)(39,41)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26820);