Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#11893 |
| 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 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,18,24,20,26,19,25)(16,27,30,43,32,44,31,41)(17,33,35,23,37,42,36,22)(21,34,38,29,40,45,39,28);(6,8)(7,10)(9,13)(11,12)(14,16)(15,21)(17,34)(18,31)(19,30)(20,32)(22,41)(23,43)(24,39)(25,38)(26,40)(27,42)(28,35)(29,36)(33,44)(37,45);(8,13)(10,12)(14,17,18,35,20,37,19,36)(15,22,25,42,26,23,24,33)(16,28,30,34,32,29,31,45)(21,27,39,41,40,44,38,43)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4881);