Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(7,9)(10,11)(12,13)(14,15,19,25,20,23,18,24)(16,26,30,21,28,36,29,37)(17,31,33,22,34,39,32,38)(27,35,43,40,41,45,42,44);(6,7,8,9)(14,16)(15,21)(17,27)(18,30)(19,29)(20,28)(22,35)(23,37)(24,36)(25,26)(31,40)(32,43)(33,42)(34,41)(38,45)(39,44);(11,13)(14,17,20,34)(15,22,23,38)(16,27,28,41)(18,33,19,32)(21,35,37,45)(24,39,25,31)(26,40,36,44)(29,43,30,42);(15,23)(16,28)(22,38)(24,25)(27,41)(29,30)(31,39)(42,43)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13923);