Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C4⋊C8 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 174 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:
(1,2,5,7,4,6,8,3)(9,10,13,19,15,17,12,18)(11,21,24,16,22,29,23,31)(14,20,27,34,28,32,26,33)(25,35,38,30,36,39,37,40);(2,6)(3,7)(9,11,14,25,12,24,26,38,15,22,28,36,13,23,27,37)(10,16,20,30,18,29,33,39,17,31,32,40,19,21,34,35);(1,3,8,6,4,7,5,2)(10,17)(11,22)(18,19)(20,32)(23,24)(25,36)(33,34)(37,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,c^b=cg,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36244);