Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133362 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #52085 | C2 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 663 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)(11,21,12,24)(13,18)(14,19)(15,20)(16,22,33,25)(17,26,34,23)(27,31)(28,36)(29,37)(30,38)(32,39,35,40);(2,5)(3,7)(9,11,15,12)(10,16,20,33)(13,25,28,22)(14,26,30,23)(17,31,34,37)(18,24,36,21)(19,32,38,35)(27,39,29,40);(9,12,15,11)(10,17,20,34)(13,25,28,22)(14,23,30,26)(16,31,33,37)(18,35,36,32)(19,21,38,24)(27,39,29,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,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, 40785);