Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×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×#102 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 363 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10,13,20,16,22,15,18)(11,23,25,34,28,39,27,38)(12,21,29,37,30,36,14,19)(17,31,33,40,35,26,32,24);(2,5,7,6)(9,11,14,26,15,27,30,40,16,28,29,31,13,25,12,24)(10,17,21,34,20,33,37,39,22,35,36,38,18,32,19,23);(10,18,22,20)(17,32,35,33)(19,36,37,21)(23,38,39,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,h^a=hi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35797);