Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1361 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7112 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 48 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)(3,15,17,45,20,47,34,48)(4,21)(5,12)(6,31)(7,33)(8,14)(9,16,35,49,38,51,39,57)(10,24)(11,30)(13,29)(18,40,28,46,32,63,19,52)(22,37,50,36,55,62,61,64)(23,44)(25,42)(26,60)(27,54)(41,58)(43,56)(53,59);(1,3)(2,9)(4,22)(5,28)(6,32)(7,34)(8,20)(10,40)(11,37)(12,36)(13,35)(14,38)(15,41)(16,25)(17,29)(18,30)(19,23)(21,48)(24,57)(26,61)(27,55)(31,62)(33,39)(42,46)(43,63)(44,64)(45,54)(47,60)(49,56)(50,58)(51,59)(52,53)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,d^2=g,e^2=hi,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1114);