Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#107 |
| prime radical | C23 | C4×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 126 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:
(1,2,4,3,5,6,8,7)(9,10,12,17,14,19,28,35,15,20,29,36,13,18,27,34)(11,21,22,30,23,37,39,32,25,38,40,31,24,33,26,16);(9,11,13,23,15,25,14,24)(10,16,18,30,20,32,19,31)(12,26,27,22,29,39,28,40)(17,33,34,21,36,37,35,38);(1,3,4,6,5,7,8,2)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cgh,d^b=dg,e^b=eg,g^b=gi,h^b=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36385);