Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1332 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)(10,11,14,19,16,21,15,20)(12,17,23,32,25,34,24,33)(13,26,30,18,29,37,28,36)(22,38,27,31,40,35,39,41);(7,9)(10,12)(11,17)(13,27)(14,23)(15,24)(16,25)(18,35)(19,32)(20,33)(21,34)(22,28)(26,31)(29,39)(30,40)(36,38)(37,41);(10,13)(11,18)(12,22)(14,28)(15,30)(16,29)(17,31)(19,26)(20,37)(21,36)(23,39)(24,27)(25,40)(32,38)(33,35)(34,41);(7,9)(10,14,16,15)(11,19,21,20)(12,23,25,24)(13,28,29,30)(17,32,34,33)(18,26,36,37)(22,39,40,27)(31,38,41,35)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17391);