Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93772 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7076 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | #2113 | C22 |
| prime radical | #13313 | C2 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ #2113 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 28 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:
(1,2,4,11,8,16,27,49)(3,17,19,43,22,37,57,61)(5,28,29,59,31,13,24,46)(6,32,25,45,33,12,50,51)(7,14,26,47,30,15,23,48)(9,34,35,21,38,56,62,58)(10,39,40,20,42,55,44,53)(18,36,52,60,54,41,63,64);(2,9)(3,18)(4,23)(5,6)(7,30)(10,13)(11,43)(12,36)(14,37)(15,17)(16,38)(19,55)(20,21)(22,54)(25,50)(26,27)(28,42)(31,33)(32,41)(34,52)(35,47)(39,57)(40,51)(44,45)(46,64)(48,62)(49,61)(53,58)(56,63)(59,60)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 970);