Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22);(2,5)(6,8)(9,11)(10,14)(12,20,21,17)(13,18)(15,24,19,22)(16,23);(3,7)(6,8)(9,12)(10,15,16,19)(11,17)(13,21)(14,22,23,24)(18,20);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=e,b^a=be,c^a=cf,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289966);