Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11369 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #399 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,20,23,33)(13,17)(14,18)(15,19)(16,22,29,26)(21,36,38,35)(24,31)(25,32)(27,34)(28,39,30,37)(40,41);(7,9)(10,12,14,26,15,23,27,22)(11,16,18,33,19,29,34,20)(13,21,24,39,25,38,40,37)(17,30,31,35,32,28,41,36);(7,9)(10,13,14,24,15,25,27,40)(11,17,18,31,19,32,34,41)(12,21,26,39,23,38,22,37)(16,28,33,36,29,30,20,35)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4357);