Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33671 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
554 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)(14,15)(16,26)(17,23)(18,24)(19,22)(20,34)(21,28)(25,33)(27,41)(29,40)(30,36)(31,38)(32,39)(35,44)(37,43)(42,45);(7,10)(11,13)(14,16,18,28)(15,21,24,26)(17,27,20,30)(19,29,32,44)(22,35,39,40)(23,36,34,41)(25,37,38,42)(31,43,33,45);(15,22)(17,20)(19,33)(21,35)(23,25)(24,39)(26,40)(27,30)(29,45)(31,32)(34,38)(36,37)(41,42)(43,44);(6,8,9,12)(7,11,10,13)(14,17,18,20)(15,23,24,34)(16,27,28,30)(19,31,32,33)(21,36,26,41)(22,38,39,25)(29,43,44,45)(35,42,40,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26659);