Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,3)(5,6,7,8)(9,10,12,11);(1,2,4,3)(9,11,12,10)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(1,3,4,2)(10,11)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,f^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,h^b=hi,f^c=fh,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56132);