Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1365 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
132 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,21,28,35,13,31,29,39,25,40,27,38,16,22,30,36,33,20,11,23,15,19,24,34,32,37,26,18);(1,3,4,8)(2,5,6,7)(10,17,22,36)(11,24)(12,27)(13,15)(14,32)(18,20,38,21)(19,39,31,34)(23,37,35,40)(25,33)(26,30)(28,29)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fhi,d^2=g,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=def,e^b=ei,f^b=fh,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1136);