Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,23,22,19)(25,26,31,34,29,33,30,39)(27,35,37,40,36,32,28,38);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,21)(19,23)(25,27,30,28,29,36,31,37)(26,32,39,40,33,35,34,38);(2,5)(6,8)(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24)(25,28,30,36,29,37,31,27)(26,32,34,38,33,35,39,40);(1,3,4,7)(2,6,5,8)(10,14)(15,22)(16,21)(17,24)(26,33)(27,36)(28,37)(34,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126405);