Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94533 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7344 |
| commutator | C4⋊Q8 | C2×C8 |
| Frattini, cosocle | #1960 | C22 |
| prime radical | #3544 | C4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1960 ◁ G | |
| LECS | G ▷ #1960 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 26 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)(3,13,15,45,16,35,23,59)(4,18,20,55)(6,12,26,42)(7,28,30,41)(8,31,32,48,34,49,61,53)(9,24,36,22)(11,25,40,27)(14,43,47,62,19,33,56,38)(17,52,54,60,51,46,50,64)(21,44,58,39)(29,37,57,63);(1,3)(2,8)(4,19)(5,23)(6,16)(7,17)(9,35)(10,38)(11,43)(12,34)(13,37)(14,21)(15,26)(18,46)(20,47)(22,53)(24,49)(25,54)(27,51)(28,33)(29,48)(30,50)(31,57)(32,40)(36,60)(39,45)(41,61)(42,62)(44,52)(55,59)(56,58)(63,64)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=dehi,e^b=eh,f^b=fi,g^b=gh,d^c=dghi,e^c=eh,g^c=gi,e^d=ehi,g^d=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1729);