Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143610 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33771 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(2,6)(3,4)(7,8)(9,10)(11,18)(12,17)(13,31)(14,22)(15,21)(16,23)(19,28)(20,30)(24,39)(25,36)(26,34)(27,37)(29,33)(32,35)(38,40);(1,2)(3,8)(4,6)(5,7)(10,17)(11,24)(12,28)(13,30)(14,15)(18,34)(19,21)(20,22)(23,31)(25,29)(26,32)(27,40)(33,35)(36,37);(1,2)(3,6)(4,8)(5,7)(9,11,16,27)(10,18,23,37)(12,29,13,32)(14,26,15,25)(17,33,31,35)(19,38,20,39)(21,36,22,34)(24,28,40,30)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=gh,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fhi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fhi,g^c=gi,h^c=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50807);