Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#61488 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#529 ◁ G | |
1788 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)(7,10)(11,13)(14,15)(16,25)(17,29)(18,23)(19,24)(20,33)(21,28)(22,30)(26,41)(27,37)(31,38)(32,36)(34,39)(35,42)(40,44)(43,45);(6,7)(8,11)(9,10)(12,13)(14,16)(15,21)(17,26)(18,32)(19,28)(20,27)(22,35)(23,37)(24,25)(29,45)(30,44)(31,43)(33,36)(34,42)(38,41)(39,40);(14,17,19,31)(15,22,24,39)(16,26,28,43)(18,30,20,34)(21,35,25,40)(23,38,33,29)(27,42,32,44)(36,45,37,41);(7,10)(11,13)(14,18,19,20)(15,23,24,33)(16,27,28,32)(17,30,31,34)(21,36,25,37)(22,38,39,29)(26,42,43,44)(35,45,40,41);(6,8,9,12)(7,11,10,13)(16,28)(21,25)(26,43)(27,32)(35,40)(36,37)(41,45)(42,44)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 54476);