Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(9,10,13,19,14,18,26,33,15,20,27,34,12,16,25,29)(11,17,23,31,24,30,37,39,21,32,35,40,22,28,36,38);(1,2,3,6,4,5,7,8)(10,16,20,18)(11,21)(17,30,32,28)(19,29,34,33)(22,24)(23,35)(31,39,40,38)(36,37);(9,11,13,23,14,24,26,37,15,21,27,35,12,22,25,36)(10,17,19,31,18,30,33,39,20,32,34,40,16,28,29,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=di,f^a=fgi,c^b=ci,d^b=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35701);