Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
702 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)(9,10,13,19,14,18,28,34,15,20,27,35,12,17,26,33)(11,21,24,36,25,31,38,40,22,32,37,39,23,16,29,30);(2,4)(5,6,7,8)(9,11)(10,16,20,31)(12,23)(13,24)(14,25)(15,22)(17,32,18,21)(19,30,35,40)(26,29)(27,37)(28,38)(33,39,34,36);(10,17,20,18)(11,22)(13,27)(16,21,31,32)(19,34,35,33)(23,25)(26,28)(30,39,40,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ch,d^a=di,f^a=fi,g^a=gi,c^b=ci,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53505);