Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60666 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,18,20)(16,24,26,37)(17,22,28,33)(19,31,30,23)(21,34,32,27)(25,38,39,45)(29,42,41,36)(35,44,43,40);(6,8,9,13)(7,10,11,12)(15,20)(21,32)(22,33)(23,31)(24,37)(35,43)(36,42)(38,45);(14,16,19,27,18,26,30,34)(15,21,23,24,20,32,31,37)(17,25,29,40,28,39,41,44)(22,35,36,38,33,43,42,45);(14,17)(15,22)(16,25)(18,28)(19,29)(20,33)(21,35)(23,36)(24,38)(26,39)(27,40)(30,41)(31,42)(32,43)(34,44)(37,45);(17,28)(22,33)(25,39)(29,41)(35,43)(36,42)(38,45)(40,44)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53654);