Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7916 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(10,11)(12,18,20,15)(13,16)(14,17)(19,24,25,22)(21,23)(26,27,29,33,30,32,38,40)(28,34,36,41,35,31,37,39);(6,7,8,9)(10,12)(11,15)(13,19)(14,20)(16,22)(17,18)(21,25)(23,24)(26,28)(27,31)(29,37)(30,35)(32,34)(33,41)(36,38)(39,40);(10,13,14,21)(11,16,17,23)(12,19,20,25)(15,22,18,24)(28,35)(31,34)(36,37)(39,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 904);