Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2054 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12,15,21)(7,16,19,20)(14,24,31,36)(18,32,35,25);(5,6,9,12,11,15,8,21)(7,17,16,30,19,34,20,33)(10,24,23,31,26,36,22,14)(13,25,28,18,29,32,27,35);(5,7,8,16,11,19,9,20)(6,13,12,27,15,29,21,28)(10,25,22,18,26,32,23,35)(14,30,24,17,31,33,36,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56138);