Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #383 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,19,21,23)(12,15)(13,17)(14,18)(16,20,24,22);(2,4)(10,15,18,17)(16,19,24,23);(6,8)(9,11,13,22,14,21,12,20)(10,16,15,19,18,24,17,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,h^a=hi,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72215774, #72216103, #72235495, … |
| maximal normal subgroup of | #72215774, #72216103, #72235495, … |
| maximal quotient of | #72215734, #72235546 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30640);