Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C4⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4⋊#95 |
| prime radical | C24 | C4⋊Q8 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,12,16,14,17,13,22)(11,18,20,15,21,23,19,24);(1,2,5,3,6,8,4,7)(9,11,13,20,14,21,12,19)(10,15,16,18,17,24,22,23);(1,3,4,2,6,7,5,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,f^b=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16089);