Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124681 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1514 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(2,4)(5,6)(7,11)(8,17,18,12)(9,13)(10,15)(14,19,20,16);(1,2,3,4)(6,11,15,13)(8,18)(12,14,17,20)(16,19);(6,12)(7,9)(8,19)(11,14)(13,20)(15,17)(16,18)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41657);