Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149774 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#496 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 70 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)(5,6,9,14)(7,12,15,19)(8,13,16,20)(10,11,17,18);(1,2,3,4)(5,7,8,10)(6,11,13,12)(9,15,16,17)(14,18,20,19);(6,12)(7,10)(9,17)(11,13)(15,16)(18,19)Polycyclic presentation:
2^9|a^2=f,b^2=d,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^c=eh,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59421);