Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8435 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| 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 | |
696 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)(6,7,11,12,10,8,9,13)(14,15,19,25,20,23,18,24)(16,26,28,39,29,21,34,36)(17,30,32,22,33,38,31,37)(27,41,42,35,43,45,44,40);(6,8,10,7)(9,13,11,12)(14,16)(15,21)(17,27)(18,28)(19,34)(20,29)(22,35)(23,26)(24,36)(25,39)(30,45)(31,42)(32,44)(33,43)(37,40)(38,41);(6,9,10,11)(7,12,8,13)(14,17,20,33)(15,22,23,37)(16,27,29,43)(18,32,19,31)(21,35,26,40)(24,38,25,30)(28,44,34,42)(36,41,39,45)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1423);