Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#120 | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 670 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:
(1,2,3,6,4,5,7,8)(9,10)(11,15)(12,22,23,32)(13,17)(14,18)(16,24,30,25)(19,34,35,39)(20,28)(21,29)(26,33)(27,36,40,37)(31,38);(2,5)(3,7)(9,11,14,21)(10,15,18,29)(12,19,23,35)(13,20,26,38)(16,27,30,40)(17,31,33,28)(22,39,32,34)(24,36,25,37);(9,12,14,23)(10,16,18,30)(11,19,21,35)(13,25,26,24)(15,27,29,40)(17,32,33,22)(20,37,38,36)(28,39,31,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=hi,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fi,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40781);