Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143719 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33580 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,5)(3,7)(6,8)(9,10)(11,22)(12,21)(13,17)(14,18)(15,20)(16,23)(19,29)(24,31)(25,33)(26,37)(27,36)(28,32)(30,35)(34,40)(38,39);(1,2,4,6)(3,8,7,5)(9,11,15,26)(10,16,20,33)(12,25,14,23)(13,28,30,24)(17,34,35,38)(18,37,21,22)(19,31,36,32)(27,39,29,40);(10,17)(11,23)(12,14)(13,29)(16,31)(18,36)(19,21)(20,35)(22,38)(24,28)(25,26)(27,30)(32,33)(34,37)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50939);