Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132067 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8456 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #405 | C23 |
| prime radical | #525 | C2×C4 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,5,10,24,38,58,60,7,12,26,40,6,11,25,39)(3,13,16,42,27,59,22,56,15,33,47,61,17,43,49,64)(4,18,20,51,52,9,32,35,19,50,23,57,21,34,54,37)(8,28,29,14,41,44,63,55,31,53,62,46,30,48,36,45);(1,3,7,15)(2,8,12,31)(4,14,19,46)(5,22,26,49)(6,27,24,17)(9,33,34,13)(10,36,40,63)(11,41,38,30)(16,25,47,58)(18,43,50,59)(20,53,23,28)(21,55,52,45)(29,60,62,39)(32,44,54,48)(35,56,37,64)(42,51,61,57);(9,34)(13,33)(18,50)(35,37)(42,61)(43,59)(51,57)(56,64)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39266);