Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,12,16,13,14,20,21)(11,17,19,24,18,15,22,23);(1,2,4,5)(3,6,7,8)(10,14)(12,20)(15,17)(19,22);(2,5)(6,8)(9,11)(10,15,14,17)(12,19)(13,18)(16,23,21,24)(20,22);(1,3,4,7)(2,6,5,8)(10,14)(12,20)(15,17)(19,22)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271252);