Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| 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 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 390 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,13,20,16,22,15,32)(11,23,25,33,28,38,27,36)(12,18,30,37,31,21,14,19)(17,26,34,29,35,40,39,24);(1,2,3,6,4,5,7,8)(9,11,14,26,15,27,31,24,16,28,30,40,13,25,12,29)(10,17,21,33,20,34,19,38,22,35,18,36,32,39,37,23);(2,5)(6,8)(9,12,13,30,16,31,15,14)(10,18,20,37,22,21,32,19)(11,24,25,26,28,29,27,40)(17,33,34,38,35,36,39,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cgh,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35788);