Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Hall | C3×C5 ⋊ 2−1+4 |
| Sylow | C5 ⋊ #1010 |
| Sylow | C3 ⋊ #4147 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D5 |
| commutator, socle, cosocle | C2×C3×C5 | C24 |
| Fitting | C2×C3×C4×C5 | C22 |
| hypercenter | C2×C4 | D3×D5 |
| hypocenter | C3×C5 | 2−1+4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
292 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5,3)(6,7,8)(10,16)(14,22)(15,24)(21,23);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23);(7,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21);(2,3)(4,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)Polycyclic presentation:
2^5,3,5|b^2=e,c^2=e,d^a=de,g^a=g^4,c^b=ce,f^b=f^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11747122, #11747123, #11748253, … |
| maximal normal subgroup of | #11747122, #11747123, #11748253, … |
| maximal quotient of | #11752460, #11752461, #11752462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1076);