Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C5 ⋊ #125 |
| Sylow | C5 ⋊ #889 |
| Sylow | C3 ⋊ #4036 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D5 |
| commutator, socle, prime radical, cosocle | C22×C3×C5 | C23 |
| Fitting | C2×C3×C4×C5 | C22 |
| hypercenter | C2×C4 | D3×D5 |
| hypocenter | C3×C5 | #125 |
| LCS | G ▷ C22×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
112 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 480 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,5,3)(6,7,8)(11,15)(13,16)(17,19,20,24)(18,21,22,23);(7,8)(9,11,12,15)(10,13,14,16);(2,3)(4,5)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,23,24,21)Polycyclic presentation:
2^5,3,5|a^2=d,b^2=e,c^2=de,c^a=cde,g^a=g^4,c^b=ce,f^b=f^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11752461, #11752462, #11752482, … |
| maximal normal subgroup of | #11752461, #11752462, #11752482, … |
| maximal quotient of | #11748571, #11748583, #11748591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 422);