Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(10,13)(11,14)(15,16)(18,21)(19,23)(22,24);(1,3)(2,5)(4,7)(6,8)(9,10,14,16,12,15,11,13)(18,22)(21,24);(3,7)(5,8)(17,18,23,24,20,22,19,21);(9,10,14,16,12,15,11,13)Polycyclic presentation:
2^9|a^2=gi,b^2=eh,c^2=fi,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 222789);