Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94453 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7318 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, cosocle | #3491 | C22 |
| prime radical | #534 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ #3491 ◁ G | |
| LECS | G ▷ #3491 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 276 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)(3,13,15,37)(4,18,19,46)(6,24,22,33)(7,12,23,34)(8,26,27,14)(9,30,31,21)(11,35,36,25)(16,44,42,62)(17,39,43,58)(20,32,49,60)(28,56,57,64)(29,48,55,41)(38,61,63,45)(40,51,53,54)(47,52,59,50);(1,3,6,16,7,17,25,45)(2,8,11,28,12,29,24,51)(4,14,20,40,21,41,50,64)(5,15,22,42,23,43,35,61)(9,13,32,38,18,39,47,44)(10,27,33,54,34,55,36,57)(19,48,49,56,30,26,52,53)(31,58,59,63,46,37,60,62)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,e^2=i,b^a=bc,c^a=cf,e^a=egh,g^a=ghi,h^a=hi,c^b=cg,d^b=dfh,f^b=fh,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
| maximal quotient of | #72267637, #72267638 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1650);