Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#9088 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
480 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)(6,7,9,12,10,8,11,13)(14,15)(16,25)(17,29,32,39)(18,33)(19,23)(20,24)(21,27)(22,31,38,30)(26,41,43,35)(28,36)(34,37)(40,42,45,44);(6,8,9,13,10,7,11,12)(14,16,18,27,20,28,19,34)(15,21,23,25,24,37,33,36)(17,26,31,44,32,43,30,42)(22,35,39,40,38,41,29,45);(14,17,20,32)(15,22,24,38)(16,26,28,43)(18,31,19,30)(21,35,37,41)(23,39,33,29)(25,40,36,45)(27,44,34,42)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2076);