Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| automorphism | Aut(G) = D3×AGL(1,ℤ/8)×AGL(1,𝔽11) (order 21120) · Inn(G) = D132 · Out(G) = C24×C5 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic132 |
| Hall | C3×C11 ⋊ Q16 |
| Sylow | C11 ⋊ Dic12 |
| Sylow | C3 ⋊ Dic44 |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 528 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,30)(22,29)(23,28)(24,27)(25,26)Polycyclic presentation:
2^4,3,11|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^10135 conjugacy classes — view the full character table.
| maximal subgroup of | #11760479, #11760481, #11760484, … |
| maximal normal subgroup of | #11760479, #11760481, #11760484, … |
| maximal quotient of | #11760638, #11760642, C2×Dic132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 67);