Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
590 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,14,10,13,12,19,11,7,15,18,8,17,16,20);(5,7,16,19,10,17,9,18,11,6,12,20,8,13,15,14);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,f^b=fgi,g^b=gi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44942);