Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33686 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
544 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)(6,7)(8,11)(9,10)(12,13)(15,20)(16,24)(18,29)(19,28)(21,35)(22,38)(25,43)(26,44)(27,42)(30,31)(32,36)(33,40)(37,39)(41,45);(7,10)(11,13)(14,15)(16,21)(17,20)(18,22)(19,23)(24,36)(25,32)(26,37)(27,33)(28,34)(29,40)(30,39)(31,41)(35,43)(38,42)(44,45);(14,16,17,25)(15,21,20,32)(18,30,27,44)(19,31,28,26)(22,39,33,45)(23,41,34,37)(24,42,43,29)(35,40,36,38);(6,8,9,12)(7,11,10,13)(15,20)(21,32)(22,33)(23,34)(35,36)(37,41)(38,40)(39,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26674);