Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,8,7)(9,10)(11,14)(12,13)(15,16);(10,12)(14,16)(18,21)(22,24);(1,2,4,3)(5,6,8,7)(11,15)(14,16)(17,18,20,21)(19,22,23,24);(1,3,4,2)(5,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24);(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 446683);