Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#7721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
444 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)(8,12,13,10)(9,11)(14,15)(16,23,25,28)(17,26)(18,21)(19,22)(20,24,29,27)(30,31,33,36,34,37,40,43)(32,35,39,42,38,41,44,45);(6,8)(7,10)(9,13)(11,12)(14,16,19,25)(15,20,22,29)(17,24,18,27)(21,28,26,23)(30,32)(31,35)(33,39)(34,38)(36,42)(37,41)(40,44)(43,45);(7,11)(10,12)(14,17,19,18)(15,21,22,26)(16,24,25,27)(20,28,29,23)(32,38)(35,41)(39,44)(42,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 709);