Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12664 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C16 | D4 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C16 ◁ C22×C5×C16 ◁ G | |
334 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)(10,11,15,20,16,21,33,30,17,22,34,29,14,19,28,13)(12,23,26,24,25,37,40,39,27,38,32,41,31,18,35,36);(6,7,8,9)(10,12)(11,18)(13,24)(14,31)(15,32)(16,25)(17,27)(19,38)(20,39)(21,23)(22,37)(26,34)(28,40)(29,36)(30,41)(33,35);(10,13,28,19,14,29,34,22,17,30,33,21,16,20,15,11)(12,24,40,38,31,36,26,37,27,41,35,23,25,39,32,18)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5652);