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 | #382 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1774 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)(11,21)(12,26,28,33)(13,19)(14,18)(15,20)(16,23)(17,27,34,29)(22,30,37,40)(24,32)(25,31)(35,36,39,38);(2,4)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,22,29,38,28,37,27,36)(17,30,26,39,34,40,33,35);(6,8)(9,12)(10,17,20,34)(11,22)(13,27)(14,29)(15,28)(16,30,32,40)(18,33,19,26)(21,35,31,39)(23,36)(24,38)(25,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,g^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30639);