Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133293 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 821 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(5,8)(9,10,14,19,15,20,12,17)(11,21,24,39,25,36,22,38)(13,27,26,37,29,18,30,35)(16,31,32,28,33,40,34,23);(1,2)(3,5)(4,6)(7,8)(9,11,15,25)(10,16,20,33)(12,24,14,22)(13,28,29,23)(17,34,19,32)(18,36,27,21)(26,31,30,40)(35,38,37,39);(1,3)(2,5)(4,7)(6,8)(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,26,29,30)(16,32,33,34)(18,35,27,37)(21,38,36,39)(23,31,28,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgi,d^c=dh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40497);