Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94449 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7313 |
| commutator | #117 | C42 |
| Frattini, cosocle | #3510 | C22 |
| prime radical | #535 | D4 |
| LCS | G ▷ #117 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3510 ◁ G | |
| LECS | G ▷ #3510 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 2 generators on 64 points:
(1,2,5,10,7,11,26,41)(3,13,15,43,17,36,48,60)(4,18,20,51,22,52,24,59)(6,27,29,40,30,12,25,42)(8,31,32,54,33,55,61,47)(9,35,37,57,38,56,63,21)(14,45,46,39,19,34,53,62)(16,44,23,58,28,50,49,64);(1,3)(2,8,11,33)(4,19)(5,23)(6,28,30,16)(7,17)(9,36,38,13)(10,39)(12,34)(14,22)(15,25,48,29)(18,50)(20,55)(21,47,57,54)(24,31)(26,49)(27,45)(32,40,61,42)(35,53,56,46)(37,58)(41,62)(43,51,60,59)(44,52)(63,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,d^b=dfh,f^b=fh,h^b=hi,d^c=dhi,g^c=gi,e^d=eh,f^d=fi,g^d=gi,f^e=fi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1647);