Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143235 |
| 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 | #28467 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,4,7)(3,5,8,6)(9,10,14,18)(11,20,24,31)(12,16,28,34)(13,29,27,19)(15,22,32,21)(17,36,35,26)(23,40,30,37)(25,38,39,33);(2,5)(3,8)(6,7)(9,11)(10,15)(12,25)(13,30)(14,24)(16,20)(17,37)(18,32)(19,38)(21,26)(22,36)(23,27)(28,39)(29,33)(31,34)(35,40);(10,16)(12,26)(13,27)(15,31)(17,19)(18,34)(20,32)(21,25)(22,39)(23,30)(28,36)(29,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50610);