Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140281 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #468 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5)(4,6)(7,8)(9,10,15,22,17,24,16,23)(11,25,29,35,30,40,12,31)(13,21,14,20,26,39,32,38)(18,28,36,27,19,34,37,33);(2,5)(6,7)(9,11,26,27,15,12,32,28,17,30,13,33,16,29,14,34)(10,18,20,40,23,36,21,31,24,19,38,25,22,37,39,35);(2,6)(4,8)(5,7)(9,12,32,27,16,11,26,34,17,29,14,33,15,30,13,28)(10,19,39,35,22,36,38,25,24,18,21,31,23,37,20,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=dh,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=egh,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72309488, #72309495, #72309504, … |
| maximal normal subgroup of | #72309488, #72309495, #72309504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47494);