Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11934 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C23×C5 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
306 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)(7,10)(8,12)(11,13)(14,15,19,25,20,26,35,42,21,27,33,40,17,23,18,24)(16,28,31,39,29,41,30,43,32,44,34,22,36,38,45,37);(6,7,9,11)(8,13,12,10)(14,16)(15,22)(17,29)(18,34)(19,30)(20,36)(21,32)(23,37)(24,41)(25,28)(26,43)(27,39)(31,35)(33,45)(38,42)(40,44);(14,17,21,20)(15,23,27,26)(16,29,32,36)(18,33,35,19)(22,37,39,43)(24,40,42,25)(28,41,44,38)(30,34,45,31)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4922);