Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7310 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1304 |
| commutator | C24 | C4×C8 |
| Frattini, cosocle | C22×#131 | C22 |
| socle | C2 | #7032 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#131 ▷ C22×C4 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,6,12,24,37,7,11,25,36,26,38,51,60)(3,13,15,39,17,40,44,61,16,31,43,53,45,56,63,64)(4,18,20,35,22,46,48,59,21,9,23,32,49,33,50,57)(8,27,28,47,30,52,54,14,29,41,34,42,55,62,58,19);(1,3)(2,8)(4,19,21,14)(5,15)(6,17)(7,16)(9,31,18,13)(10,34)(11,29)(12,30)(20,27,23,41)(22,47,49,42)(24,44)(25,43)(26,45)(28,36)(32,39,35,53)(33,56,46,40)(37,58)(38,55)(48,52,50,62)(51,63)(54,60)(57,61,59,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=h,h^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fi,c^b=cg,d^b=df,g^b=gi,d^c=di,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 676);