Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11018 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1270 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)(10,11,15,19,16,21,14,20)(12,22,24,17,25,32,30,33)(13,18,28,35,29,36,27,37)(23,34,38,31,39,41,26,40);(10,12)(11,17)(13,26)(14,30)(15,24)(16,25)(18,34)(19,32)(20,22)(21,33)(23,28)(27,39)(29,38)(31,35)(36,41)(37,40);(7,9)(10,13)(11,18)(12,23)(14,27)(15,28)(16,29)(17,31)(19,35)(20,37)(21,36)(22,34)(24,38)(25,39)(26,30)(32,41)(33,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4006);