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 | |
187 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 187 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:
(1,2,4,12)(3,17,19,43)(5,30,31,60)(6,13,27,49)(7,15,24,46)(8,34,28,59)(9,16,29,47)(10,35,36,22)(11,40,42,20)(14,26,45,33)(18,48,51,64)(21,39,55,62)(23,50,25,56)(32,41,58,44)(37,54,63,52)(38,57,61,53);(2,10)(3,18)(4,24)(5,6)(7,8)(11,14)(12,43)(13,48)(15,50)(16,39)(17,34)(19,40)(20,22)(21,52)(23,53)(25,54)(27,58)(28,29)(30,37)(32,33)(35,57)(36,46)(38,41)(42,49)(44,63)(45,61)(47,56)(51,55)(59,62)(60,64)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,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, 968);