Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21756 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C5×#41 ◁ G | |
1254 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)(8,12)(9,11)(10,13)(14,15,18,23,19,22,31,36)(16,20,26,34,25,33,41,44)(17,27,28,37,29,21,30,35)(24,38,40,45,39,32,42,43);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,24)(18,26)(19,25)(21,32)(22,33)(23,34)(27,38)(28,40)(29,39)(30,42)(31,41)(35,43)(36,44)(37,45);(8,13)(10,12)(14,17,19,29)(15,21,22,27)(16,24,25,39)(18,30,31,28)(20,32,33,38)(23,37,36,35)(26,42,41,40)(34,45,44,43);(7,11)(10,12)(16,25)(20,33)(24,39)(26,41)(32,38)(34,44)(40,42)(43,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14744);