Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#711 |
| Fitting | C2×C3×C5×D4 ⋊ C2 |
| Hall | C3×C5 ⋊ C2×#41 |
| Sylow | C5 ⋊ C2×C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C22×D5 |
| commutator, cosocle | C2×C5 | C24×C3 |
| Frattini | C2 | C23×C3×D5 |
| Fitting | C2×C3×C5×D4 | C2 |
| socle | C22×C3×C5 | C23 |
| hypercenter | C2×C3×D4 | D5 |
| hypocenter | C5 | C2×C3×#41 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×D4 | |
328 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2)(3,4,5)(6,7,9,10,8)(12,16)(13,17);(11,13,14,17)(12,15,16,18);(7,8)(9,10)(11,12)(13,15)(14,16)(17,18);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | #11748951, #11748961, #11749024, … |
| maximal normal subgroup of | #11748951, #11748961, #11749024, … |
| maximal quotient of | C2×C3×#4023, C2×C3×#4024, C2×C3×#4025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1140);