Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,10,11,12)(14,15,18,23,19,22,31,36)(16,24,27,37,26,20,30,33)(17,21,29,34,28,35,41,45)(25,32,40,43,39,38,42,44);(6,7)(8,12)(9,11)(10,13)(14,16)(15,20)(17,25)(18,30)(19,26)(21,32)(22,24)(23,37)(27,31)(28,39)(29,42)(33,36)(34,44)(35,38)(40,41)(43,45);(14,17,19,28)(15,21,22,35)(16,25,26,39)(18,29,31,41)(20,32,24,38)(23,34,36,45)(27,40,30,42)(33,43,37,44)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1337);