Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8503 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3554 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
672 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)(6,7,10,13,11,15,9,14)(8,16,17,12,18,20,19,21)(22,23)(24,32,35,28)(25,36,38,45)(26,31)(27,30)(29,37,43,39)(33,48)(34,47,52,42)(40,44)(41,49)(46,51)(50,53);(6,8)(7,12)(9,19)(10,17)(11,18)(13,20)(14,16)(15,21)(22,24)(23,28)(25,33)(26,34)(27,35)(29,41)(30,32)(31,42)(36,53)(37,49)(38,50)(39,51)(40,52)(43,46)(44,47)(45,48);(7,13)(9,10)(12,20)(14,15)(16,21)(17,19)(22,25,27,38)(23,29,30,43)(24,33,35,50)(26,39,40,37)(28,41,32,46)(31,45,44,36)(34,51,52,49)(42,48,47,53)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1491);