Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,23,22,19,20,24);(1,2)(3,4)(9,11,13,12)(10,14,15,16)(17,19,22,18)(20,24,21,23);(2,4)(17,18,22,19)(20,23,21,24);(5,7,8,6)(9,12,13,11)(10,14,15,16)(17,20,22,21)(18,23,19,24)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=ehi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266497);