Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8760 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1817 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,13,20,15,19,29,35,14,18,28,34,12,17,26,33)(11,21,24,36,25,32,38,40,22,16,27,31,23,30,37,39);(1,2,4,5)(3,6,7,8)(9,11,14,22)(10,16,18,21)(12,23,15,25)(13,27,28,24)(17,32,19,30)(20,36,34,31)(26,38,29,37)(33,39,35,40);(1,3,4,7)(2,6,5,8)(10,17,18,19)(11,22)(16,30,21,32)(20,33,34,35)(23,25)(24,27)(31,39,36,40)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,c^b=ci,f^b=fi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42475);