Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#9300 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
508 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)(6,7,8,12,10,13,9,11)(14,15,19,23,20,24,18,34)(16,25,29,36,28,40,27,21)(17,30,32,38,33,22,31,37)(26,39,43,35,42,45,41,44);(7,11)(8,9)(12,13)(14,16)(15,21,24,36)(17,26)(18,27)(19,29)(20,28)(22,35,30,44)(23,25,34,40)(31,41)(32,43)(33,42)(37,45,38,39);(7,12)(8,9)(11,13)(14,17,20,33)(15,22,24,30)(16,26,28,42)(18,32,19,31)(21,35,36,44)(23,38,34,37)(25,39,40,45)(27,43,29,41)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2288);