Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,20,23,16)(13,17)(14,19)(15,18)(21,38,39,30)(22,36,27,31)(24,41,40,32)(25,34)(26,33)(28,37)(29,35);(10,12)(11,16)(13,24)(14,27)(15,23)(17,32)(18,20)(19,36)(21,25)(22,29)(26,40)(28,39)(30,34)(31,35)(33,41)(37,38);(7,9)(10,13,15,26)(11,17,18,33)(12,21,23,39)(14,28,29,25)(16,30,20,38)(19,37,35,34)(22,40,27,24)(31,41,36,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4001);