Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143438 |
| 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 | C23 | #459 |
| 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 | |
418 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(2,5)(3,7)(6,8)(9,10,17,24)(11,25,31,37)(12,15,32,13)(14,23,16,21)(18,29,26,27)(19,22,34,20)(28,39,30,40)(33,36,35,38);(3,7)(5,8)(9,11,17,31)(10,18,24,26)(12,25,32,37)(13,33,15,35)(14,30,16,28)(19,29,34,27)(20,39,22,40)(21,38,23,36);(1,2)(3,8)(4,6)(5,7)(9,12,17,32)(10,19,24,34)(11,26,31,18)(13,23,15,21)(14,22,16,20)(25,33,37,35)(27,40,29,39)(28,36,30,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50641);