Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3560 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,15,17,12)(9,11)(10,14)(13,18)(16,20,21,19)(22,23,26,31,27,30,41,45)(24,32,35,46,34,28,39,43)(25,36,38,47,37,29,40,44)(33,48,50,53,49,42,51,52);(7,11)(9,18)(12,19)(13,14)(15,20)(16,21)(22,24)(23,28)(25,33)(26,39)(27,34)(29,42)(30,32)(31,46)(35,41)(36,48)(37,49)(38,51)(40,50)(43,45)(44,53)(47,52);(6,8)(7,12)(9,16)(10,17)(11,19)(13,20)(14,15)(18,21)(22,25)(23,29)(24,33)(26,40)(27,37)(28,42)(30,36)(31,47)(32,48)(34,49)(35,51)(38,41)(39,50)(43,53)(44,45)(46,52)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1477);