Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C82 |
| Frattini, cosocle | C2×C4×C22⋊C4 | C22 |
| socle | C23 | #342 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C22⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 562 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(17,18)(19,23,24,21)(20,22);(9,10,11,13,12,14,15,16)(18,21)(22,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=h,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, 478);