Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#12682 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
930 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)(10,11,13,18,14,19,25,33,15,17,26,30,27,31,38,40)(12,20,22,32,23,34,36,41,21,16,24,28,35,29,37,39);(10,12)(11,16)(13,24)(14,23)(15,21)(17,20)(18,32)(19,29)(22,26)(25,37)(27,35)(28,30)(31,34)(33,41)(36,38)(39,40);(7,9)(12,21)(16,20)(22,24)(23,35)(28,32)(29,34)(36,37)(39,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5670);