Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×C4×AGL(1,ℤ/8) (order 256) · Inn(G) = D4 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C5 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C3×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C3×C4×C5 ◁ G | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,24,23,22,21,20,19,18);(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 88);