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 | #3147 | C22 |
| 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 | |
403 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 403 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,20)(12,26)(13,18)(14,32)(15,19)(16,23)(17,30)(21,38)(22,37)(24,34)(25,40)(27,29)(28,35)(31,36)(33,39);(1,2,4,6)(3,8,7,5)(9,10)(11,21)(12,27)(13,32)(14,18)(15,19)(16,24)(17,31)(20,28)(22,35)(23,40)(25,33)(26,29)(30,36)(34,39)(37,38);(9,11,15,22)(10,16,19,33)(12,28,31,38)(13,25,14,24)(17,34,27,40)(18,35,32,21)(20,36,37,26)(23,29,39,30)Polycyclic presentation:
2^9|b^2=g,c^2=i,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, 50940);