Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#20587 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C22×C5×2−1+4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5×Q8 ◁ G | |
3634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,20,23,16)(13,17)(14,18)(15,19)(21,34,37,28)(22,35,39,29)(24,30)(25,31)(26,32)(27,33)(36,41,38,40);(10,12)(11,16)(13,21)(14,22)(15,23)(17,28)(18,29)(19,20)(24,36)(25,37)(26,38)(27,39)(30,40)(31,34)(32,41)(33,35);(7,9)(10,13,15,25)(11,17,19,31)(12,21,23,37)(14,26,27,24)(16,28,20,34)(18,32,33,30)(22,38,39,36)(29,41,35,40);(10,14,15,27)(11,18,19,33)(12,22,23,39)(13,24,25,26)(16,29,20,35)(17,30,31,32)(21,36,37,38)(28,40,34,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13575);