Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #394 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,22)(13,27,30,18)(14,21)(15,19)(16,20)(17,24)(23,35,38,40)(25,33)(26,34)(28,32,39,41)(29,36,31,37);(7,9)(10,12)(11,17)(13,28)(14,26)(15,24)(16,25)(18,35)(19,33)(20,34)(21,22)(23,29)(27,40)(30,39)(31,38)(32,36)(37,41);(10,13)(11,18)(12,23)(14,29)(15,31)(16,30)(17,32)(19,36)(20,27)(21,37)(22,35)(24,28)(25,38)(26,39)(33,40)(34,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4026);