Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(9,10)(11,16)(12,15)(13,14)(18,21)(19,23);(2,4)(9,11,14,12)(10,15,13,16)(17,18,20,21)(19,22,23,24);(1,2,3,4)(9,11,14,12)(10,16,13,15)(17,18)(19,22)(20,21)(23,24);(9,12,14,11)(10,15,13,16)(17,19)(18,22)(20,23)(21,24);(9,13,14,10)(11,15,12,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 439047);