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 | #19689 | 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 | |
392 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 392 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)(12,29)(13,18)(14,19)(15,20)(16,22)(17,24)(21,31)(25,32)(26,33)(27,34)(28,38)(30,37)(35,39)(36,40);(1,2)(3,8)(4,6)(5,7)(9,11,16,28)(10,17,22,36)(12,26,31,39)(13,27,15,25)(14,24,30,40)(18,33,20,35)(19,32,37,34)(21,38,29,23);(1,2)(3,6)(4,8)(5,7)(10,18)(11,24)(12,14)(13,15)(17,32)(19,37)(20,22)(23,35)(25,26)(27,39)(28,40)(30,31)(33,38)(34,36)Polycyclic presentation:
2^9|b^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, 50824);