Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×H(5,𝔽2) | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
483 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 483 conjugacy classes of subgroups — too many to draw. Only its 72 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)(5,6,8,12,10,7,14,16)(9,17,19,15,20,13,18,11);(1,2,3,4);(5,7)(6,10)(8,15)(9,13)(11,14)(12,18)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=f,b^2=d,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,f^c=fgi,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59405);