Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2206 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13130 |
| prime radical | C2×#508 | C22 |
| 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 | |
208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 208 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:
(1,2,6,11,8,14,4,9)(3,15,18,47,20,49,16,45)(5,23,26,58,28,59,21,46)(7,29,25,13,31,39,22,44)(10,34,35,52,37,50,32,17)(12,40,38,62,42,57,33,27)(19,54,53,41,55,64,30,48)(24,60,61,36,51,43,56,63);(2,9,14,11)(10,32,37,35)(12,33,42,38)(13,29,44,39)(15,45,49,47)(23,46,59,58)(36,60,63,43)(41,54,48,64);(1,3,4,16,8,20,6,18)(2,10,11,35,14,37,9,32)(5,24,21,56,28,51,26,61)(7,30,22,55,31,53,25,19)(12,41,38,64,42,48,33,54)(13,43,39,63,44,60,29,36)(15,46,47,23,49,58,45,59)(17,40,50,27,52,57,34,62)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=d,d^2=i,e^2=gh,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=egi,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, 60019);