Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7,9,13,10,12,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,32,35,28)(25,29)(26,30)(27,31)(33,46,49,40)(34,47,51,41)(36,42)(37,43)(38,44)(39,45)(48,53,50,52);(6,8)(7,11)(9,15)(10,14)(12,17)(13,18)(16,20)(19,21)(22,24,27,35)(23,28,31,32)(25,33,37,49)(26,34,39,51)(29,40,43,46)(30,41,45,47)(36,48,38,50)(42,52,44,53);(8,14)(11,17)(15,20)(18,21)(22,25,27,37)(23,29,31,43)(24,33,35,49)(26,38,39,36)(28,40,32,46)(30,44,45,42)(34,50,51,48)(41,53,47,52);(7,12)(11,17)(13,19)(18,21)(22,26,27,39)(23,30,31,45)(24,34,35,51)(25,36,37,38)(28,41,32,47)(29,42,43,44)(33,48,49,50)(40,52,46,53)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10398);