Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13709 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2226 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13111 |
| prime radical | C23 | #510 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 165 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) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,9,8,14,7,11)(3,15,18,47,20,49,16,45)(5,23,21,55,27,57,26,46)(6,28,22,44,31,39,30,13)(10,34,35,17,37,51,32,52)(12,40,33,61,42,60,38,25)(19,53,50,64,54,62,29,41)(24,48,58,36,59,43,56,63);(2,9,14,11)(10,32,37,35)(12,33,42,38)(13,28,44,39)(15,45,49,47)(23,55,57,46)(36,48,63,43)(41,62,64,53);(1,3,4,16,8,20,7,18)(2,10,11,35,14,37,9,32)(5,24,21,56,27,59,26,58)(6,29,22,54,31,50,30,19)(12,41,38,53,42,64,33,62)(13,43,39,63,44,48,28,36)(15,46,47,57,49,55,45,23)(17,40,34,61,52,60,51,25)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=d,d^2=i,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=eg,f^c=fi,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59999);