Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C4⋊C4 |
| prime radical | C23 | C8×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,13)(3,10)(8,15,16,14)(11,12)(17,18,20,24,21,25,28,31,22,19,23,26,29,27,30,32);(1,2,5,7,6,9,4,13)(3,8,12,14,10,16,11,15);(1,3,4,11,6,10,5,12)(2,8,13,15,9,16,7,14)(17,19,20,26,21,27,28,32,22,18,23,24,29,25,30,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=cg,d^b=dg,e^b=eg176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36392);