Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C5×#17352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
1162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,8,10,9,11,12,13)(14,15)(16,24,27,20)(17,21)(18,22)(19,23)(25,38,41,32)(26,39,43,33)(28,34)(29,35)(30,36)(31,37)(40,45,42,44);(14,16)(15,20)(17,25)(18,26)(19,27)(21,32)(22,33)(23,24)(28,40)(29,41)(30,42)(31,43)(34,44)(35,38)(36,45)(37,39);(14,17,19,29)(15,21,23,35)(16,25,27,41)(18,30,31,28)(20,32,24,38)(22,36,37,34)(26,42,43,40)(33,45,39,44);(14,18,19,31)(15,22,23,37)(16,26,27,43)(17,28,29,30)(20,33,24,39)(21,34,35,36)(25,40,41,42)(32,44,38,45)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10340);