Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C22 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C22⋊C8 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,9)(8,13)(10,15)(17,18,20,25,23,26,22,21)(19,24,27,31,30,32,29,28);(1,2,3,7,5,8,10,14,4,6,9,12,11,13,15,16)(17,19,21,28,22,29,26,32,23,30,25,31,20,27,18,24);(17,18,20,25,23,26,22,21)(19,24,27,31,30,32,29,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,e^2=gi,f^2=h,h^2=i,b^a=bd,f^a=fg,d^b=dg176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35627);