Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 | C2×#127 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,17)(12,20)(15,18)(16,22)(19,23)(25,26,30,28,29,32,38,37)(27,31,35,36,33,39,40,34);(2,5)(6,8)(9,10,12,14,13,16,20,22)(11,18,17,24,19,15,23,21)(25,27,29,33)(26,31,32,39)(28,36,37,34)(30,35,38,40);(1,3,4,7)(2,6,5,8)(9,11,12,17,13,19,20,23)(10,15,14,21,16,18,22,24)(26,32)(27,33)(28,37)(35,40);(10,16)(14,22)(15,18)(21,24)(25,28,30,32,29,37,38,26)(27,34,35,31,33,36,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=chi,d^b=dhi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 119989);