Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12008 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | D8 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#107 |
| prime radical | C23×C5 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
242 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)(6,7,8,10,9,11,12,13)(14,15)(16,25)(17,22)(18,32)(19,34)(20,24)(21,28)(23,26)(27,38)(29,44)(30,40)(31,37)(33,45)(35,42)(36,43)(39,41);(14,16,26,41,19,29,33,43,20,30,31,42,17,27,18,28)(15,21,32,38,22,35,37,40,24,36,45,44,34,39,23,25);(14,17,20,19)(15,22,24,34)(16,27,30,29)(18,31,33,26)(21,35,36,39)(23,32,37,45)(25,38,40,44)(28,42,43,41)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4996);