Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7,10,13,11,14,9,19)(8,15,18,20,16,21,17,12)(22,23)(24,33,36,44)(25,37)(26,31)(27,41)(28,32)(29,35,45,40)(30,38)(34,49,52,43)(39,46)(42,47)(48,50,53,51);(6,8,11,16)(7,12)(9,17,10,18)(13,15)(14,20)(19,21)(22,24,26,40,28,36,27,35)(23,29,31,44,32,45,41,33)(25,34,38,51,39,52,42,50)(30,43,46,53,47,49,37,48);(8,16)(12,20)(15,21)(17,18)(22,25)(23,30)(24,34)(26,38)(27,42)(28,39)(29,43)(31,46)(32,47)(33,48)(35,50)(36,52)(37,41)(40,51)(44,53)(45,49)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2302);