Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9312 |
| 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 | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,8,10,9,12,13,11)(14,15)(16,25,28,36)(17,29)(18,23)(19,33)(20,24)(21,27,37,32)(22,30)(26,41,44,35)(31,38)(34,39)(40,42,45,43);(7,10,12,11)(14,16,18,32,20,28,19,27)(15,21,23,36,24,37,33,25)(17,26,30,43,31,44,34,42)(22,35,38,45,39,41,29,40);(14,17)(15,22)(16,26)(18,30)(19,34)(20,31)(21,35)(23,38)(24,39)(25,40)(27,42)(28,44)(29,33)(32,43)(36,45)(37,41)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2300);