Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #390 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(10,14,11,15)(18,21)(22,23);(1,3,5,4)(2,6,7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(1,4,5,3)(2,6,7,8)(9,11,13,10)(12,14,16,15)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=dei,c^2=dei,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=eh,f^c=fi,g^c=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28933);