Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#60627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C5×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
1998 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)(8,11)(9,10)(12,13)(14,15,21,27)(16,23,30,36)(17,22,29,37)(18,24,32,39)(19,26,20,25)(28,38,42,45)(31,35,43,44)(33,41,34,40);(7,10)(11,13)(14,16,19,17,21,30,20,29)(15,22,25,23,27,37,26,36)(18,28,33,31,32,42,34,43)(24,35,40,38,39,44,41,45);(6,8,9,12)(7,11,10,13)(14,17,19,30,21,29,20,16)(15,23,25,37,27,36,26,22)(18,31,33,42,32,43,34,28)(24,38,40,44,39,45,41,35);(14,18)(15,24)(16,28)(17,31)(19,33)(20,34)(21,32)(22,35)(23,38)(25,40)(26,41)(27,39)(29,43)(30,42)(36,45)(37,44);(18,32)(24,39)(28,42)(31,43)(33,34)(35,44)(38,45)(40,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53615);