Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ C4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×S4 |
| commutator, cosocle | C2×C5×A4 | C22 |
| Fitting, socle | C23×C5 | C2×D3 |
| hypocenter | C5×A4 | D4 |
| LCS | G ▷ C2×C5×A4 ▷ C5×A4 | |
| UCS | C1 ◁ C2 | |
112 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,4)(5,6,8,7)(9,10,12,13,11);(1,2,3,4)(6,7,8)(10,11)(12,13)Polycyclic presentation:
2^3,3,5,2^2|a^2=c,b^a=bc,e^a=e^4,d^b=d^2,f^b=g,g^b=f,f^d=g,g^d=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | #11756746, #11756747, #11756752, … |
| maximal normal subgroup of | #11756746, #11756747, #11756752, … |
| maximal quotient of | #11751573, #11751574, #11751575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 978);