Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| 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 | |
3086 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)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24);(5,6,7,8)(17,19,22,18)(20,23,21,24);(1,3,4,2)(10,13)(14,16)(17,20,22,21)(18,23,19,24);(17,21,22,20)(18,23,19,24);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420599);