Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7338 |
| commutator | C22×D4 | C2×C8 |
| Frattini, cosocle | #1782 | C22 |
| prime radical | #3485 | C4 |
| LCS | G ▷ C22×D4 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1782 ◁ G | |
| LECS | G ▷ #1782 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 24 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)(3,13,15,44,17,46,52,64)(4,18,19,55,21,42,56,63)(6,25,23,38)(7,12,24,39)(8,29,30,53)(9,33,34,27,35,58,37,20)(11,40,43,28)(14,31,47,36)(16,45,51,62,26,54,22,57)(32,59,61,48)(41,50,60,49);(1,3)(2,8)(4,14)(5,22)(6,26)(7,17)(9,13)(10,36)(11,41)(12,32)(15,40)(16,28)(18,54)(19,29)(20,53)(21,49)(23,52)(24,51)(25,31)(27,48)(30,43)(33,50)(34,62)(35,46)(37,57)(38,61)(39,60)(42,45)(44,55)(47,58)(56,59)(63,64)Polycyclic presentation:
2^9|a^2=d,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=dei,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1718);