Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C8 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
658 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)(5,6,7,8)(9,10)(11,22)(12,18)(13,19)(14,20)(15,31)(16,21)(17,24)(23,35)(25,38)(26,33)(27,32)(28,39)(29,36)(30,37)(34,40);(2,4)(9,11,13,25,15,27,30,23,16,28,29,40,14,26,12,24)(10,17,18,33,20,34,36,39,21,35,37,32,31,38,19,22);(9,12,14,29,16,30,15,13)(10,18,20,36,21,37,31,19)(11,23,26,25,28,24,27,40)(17,32,34,22,35,33,38,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dh,f^a=fhi,h^a=hi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286901, #72286947, #72287048, … |
| maximal normal subgroup of | #72286901, #72286947, #72287048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53528);