Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143626 |
| 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 | #33780 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,23)(12,17)(13,21)(14,32)(15,20)(16,22)(18,26)(19,29)(24,36)(25,34)(27,38)(28,37)(30,39)(31,35)(33,40);(1,2)(3,8)(4,6)(5,7)(10,17)(11,24)(12,29)(13,15)(14,31)(19,21)(20,32)(22,35)(23,37)(25,26)(27,33)(28,30)(34,36)(39,40);(1,2)(3,6)(4,8)(5,7)(9,11,16,28)(10,18,22,38)(12,30,31,24)(13,27,15,26)(14,33,29,25)(17,34,35,40)(19,36,32,39)(20,37,21,23)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=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fhi,g^c=gi,h^c=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50823);