Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13002 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
528 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)(10,11,15,20,16,21,28,38,17,22,14,30,13,19,29,36)(12,23,26,34,24,37,32,35,27,40,25,39,33,41,31,18);(10,12)(11,18,22,35)(13,24)(14,31)(15,32)(16,33)(17,27)(19,34,21,39)(20,37,30,41)(23,36,40,38)(25,29)(26,28);(7,9)(10,13,17,16)(11,19,22,21)(12,24,27,33)(14,28,15,29)(18,34,35,39)(20,36,30,38)(23,37,40,41)(25,31,26,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5990);