Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(10,14)(15,16)(17,18)(19,22)(20,21)(23,24);(5,6,7,8)(9,10)(11,16)(12,15)(13,14)(18,21)(19,23);(1,2,4,3)(18,21)(22,24);(1,3,4,2)(9,11,13,12)(10,15,14,16)(17,19,20,23)(18,22,21,24);(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^b=df200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420694);