Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24);(5,6,7,8)(10,14)(15,16);(9,11,13,12)(10,15,14,16)(18,21)(19,24)(22,23);(1,3,4,2)(9,12,13,11)(10,16,14,15)(18,22)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261697);