Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143703 |
| 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 | #28225 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 457 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10,14,20)(11,22,25,36)(12,21,15,18)(13,17,29,34)(16,26,32,23)(19,27,35,28)(24,37,39,31)(30,38,33,40);(2,5)(3,8)(6,7)(9,11)(10,16)(12,26)(13,24)(14,25)(15,23)(17,33)(18,36)(19,31)(20,32)(21,22)(27,40)(28,38)(29,39)(30,34)(35,37);(10,17)(12,15)(13,28)(16,30)(18,35)(19,21)(20,34)(22,37)(23,26)(24,38)(27,29)(31,36)(32,33)(39,40)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,d^c=di,e^c=eh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50953);