Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
898 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21,24,35)(12,25)(13,18)(14,19,29,33)(15,20)(16,22,32,27)(17,26,34,28)(23,36)(30,38)(31,39)(37,40);(2,5)(3,7)(9,11,15,24)(10,16,20,32)(12,22,13,27)(14,23,29,38)(17,30,34,36)(18,35,25,21)(19,31,33,40)(26,37,28,39);(9,12,15,13)(10,17)(11,22,24,27)(14,26,29,28)(16,30)(18,33)(19,25)(20,34)(21,31)(23,37,38,39)(32,36)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fgi,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72284174, #72284199, #72284286, … |
| maximal normal subgroup of | #72284174, #72284199, #72284286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41289);