Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#11620 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C5×#121 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,10,13,11,8,9,12)(14,15,18,24,19,25,33,42,20,22,34,38,17,23,31,40)(16,26,29,43,28,37,44,41,27,39,32,45,30,21,35,36);(6,8)(7,11)(9,13)(10,12)(14,16)(15,21)(17,30)(18,32)(19,28)(20,27)(22,37)(23,39)(24,41)(25,26)(29,34)(31,44)(33,35)(36,38)(40,43)(42,45);(6,9,11,10)(7,12,8,13)(15,22)(16,27)(23,25)(24,38)(28,30)(29,32)(35,44)(40,42)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4608);