Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| 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 | |
313 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 313 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)(9,10,13,19,14,18,27,34,15,20,28,35,12,16,26,30)(11,21,23,36,24,29,38,39,25,17,31,33,22,32,37,40);(1,2,3,6,4,5,7,8)(10,16,20,18)(17,29,21,32)(19,30,35,34)(33,39,36,40);(9,11,13,23,14,24,27,38,15,25,28,31,12,22,26,37)(10,17,19,33,18,32,34,40,20,21,35,36,16,29,30,39)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,c^a=ci,d^a=di,f^a=fgi,d^b=dg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35702);