Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×Q8 | C22×C4×C5 |
| Frattini, cosocle | C22×Q8 | C23×C5 |
| prime radical | C5×#3545 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×Q8 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,14,12,17,10,16)(8,18,20,36,21,32,26,35)(9,15,24,33,25,34,23,31)(13,27,29,19,30,37,28,22);(6,8)(7,13)(9,22)(10,26)(11,20)(12,21)(14,28)(15,32)(16,29)(17,30)(18,34)(19,25)(23,37)(24,27)(31,35)(33,36);(7,14)(10,11)(13,28)(15,33)(16,17)(18,35)(20,26)(23,24)(27,37)(29,30)(31,34)(32,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5769);