Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C4×C16 |
| Frattini, cosocle | C22×M32 | C22 |
| socle | C22 | #1314 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6,7,12,8,13,16,24,9,14,17,25,18,26,29,34,10,11,15,21,19,22,27,31,20,23,28,32,30,33,35,36);(1,2,3,4)(6,11)(13,22)(14,23)(26,33)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,f^a=fi,d^b=df,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1260);