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 | C2×C4 |
| LCS | G ▷ #117 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3510 ◁ G | |
| LECS | G ▷ #3510 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
262 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 262 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:
(1,2,4,10)(3,14,16,32)(5,11,23,37)(6,12,26,9)(7,13,22,38)(8,19,29,47)(15,31,45,35)(17,44,50,30)(18,53,48,41)(20,55,25,40)(21,34,28,36)(24,39,49,43)(27,42,56,33)(46,59,51,62)(52,58,64,60)(54,63,57,61);(1,3,5,17,7,19,25,52)(2,8,11,30,13,32,40,60)(4,15,20,46,22,48,23,57)(6,18,27,54,28,45,24,51)(9,29,33,58,34,14,43,44)(10,35,37,62,38,53,55,63)(12,41,42,59,36,31,39,61)(16,49,50,21,47,56,64,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,e^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, 1646);