Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #380 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(6,8)(9,10,14,19,15,20,13,18)(11,21,25,16,24,32,23,31)(12,17,28,35,29,36,27,34)(22,37,39,30,26,40,38,33);(2,4)(9,11)(10,16)(12,26,29,22)(13,25)(14,23)(15,24)(17,33,36,30)(18,32)(19,21)(20,31)(27,38,28,39)(34,37,35,40);(5,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,34)(19,35)(20,36)(21,37)(23,38)(24,26)(25,39)(31,33)(32,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fi,g^b=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31226);