Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C4⋊C8 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 206 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 — 3 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,13,19,15,17,12,18)(11,21,23,16,25,30,22,29)(14,20,27,34,28,32,26,33)(24,35,37,31,38,40,36,39);(9,11,14,24,12,23,26,37,15,25,28,38,13,22,27,36)(10,16,20,31,18,30,33,40,17,29,32,39,19,21,34,35);(1,3,4,8,5,2,6,7)(10,17)(16,29)(18,19)(20,32)(21,30)(31,39)(33,34)(35,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,d^b=di,e^b=ei176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36235);