Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18548 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(10,14)(11,17)(12,20)(16,22)(18,19)(21,23);(9,10,12,14,13,16,20,22)(11,15,18,21,19,24,17,23);(1,2,4,6)(3,8,7,5);(3,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,21)(16,24)(17,20)(22,23)Polycyclic presentation:
2^9|b^2=e,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^c=gi,f^d=fi,g^d=gh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255255);