Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #389 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,6,7,8)(9,10,13,19,15,20,14,18)(11,21,24,32,25,31,23,16)(12,26,27,17,29,33,28,34)(22,30,37,35,38,40,36,39);(1,2)(3,4)(9,11,14,23,15,25,13,24)(10,16,18,31,20,32,19,21)(12,22,28,36,29,38,27,37)(17,30,26,39,34,40,33,35);(2,4)(9,12,13,27,15,29,14,28)(10,17,18,26,20,34,19,33)(11,22,24,37,25,38,23,36)(16,30,31,39,32,40,21,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31083);