Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125287 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | #32936 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
669 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 669 conjugacy classes of subgroups — too many to draw. Only its 98 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)(7,15,12,20)(8,18,19,17)(9,13)(10,14)(11,16);(5,7,15,6,11,17,19,13,10,12,20,14,9,18,8,16);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45156);