Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7077 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2112 | C22 |
| prime radical | C24 | Q32 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2112 ◁ G | |
| LECS | G ▷ #2112 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 120 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:
(1,2,5,12,9,16,8,15)(3,17,19,52,22,54,36,53)(4,23,26,51,29,49,28,59)(6,31,30,13,7,34,25,14)(10,37,38,55,42,61,41,18)(11,43,45,27,47,63,46,33)(20,56,32,44,21,58,35,64)(24,48,60,39,57,50,62,40);(1,3,5,19,9,22,8,36)(2,10,15,41,16,42,12,38)(4,24,28,60,29,57,26,62)(6,32,30,21,7,35,25,20)(11,44,45,56,47,64,46,58)(13,48,31,40,14,50,34,39)(17,51,53,49,54,59,52,23)(18,43,37,33,55,63,61,27)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fhi,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=defhi,e^b=eh,f^b=fh,h^b=hi,d^c=dghi,e^c=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 989);