Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of SL(2,𝔽9) |
| automorphism | Aut(G) = PΓL(2,𝔽9) (order 1440) · Inn(G) = A6 · Out(G) = C22 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C3 |
| special linear | SL(2,𝔽9) |
| symplectic | Sp(2,𝔽9) |
| special unitary | SU(2,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | A6 |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 80 points:
(1,12,3,39,41)(2,75,58,10,9)(4,47,72,46,59)(5,45,54,30,44)(6,28,21,77,61)(7,52,20,74,19)(8,63,15,70,62)(11,26,42,64,40)(13,31,48,67,57)(14,71,79,23,43)(16,69,34,35,25)(17,56,38,65,37)(18,53,24,36,49)(22,76,80,60,68)(27,66,50,33,78)(29,73,51,32,55);(1,72,50,22,6,80,42,4)(2,26,64,74,20,66,33,75)(3,32,29,77,21,63,70,39)(5,8,25,11,23,51,17,27)(7,49,15,31,10,67,55,24)(9,38,19,76,52,69,58,47)(12,54,61,48,28,14,41,36)(13,57,16,34,53,18,56,65)(30,59,46,43,71,68,60,45)(35,62,44,78,37,73,79,40)13 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 409);