Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| socle | C22 | C2×AΣL(1,𝔽16) |
| prime radical | #23406 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 671 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,16)(11,14,19,21)(13,18,23,24)(15,20,17,22);(2,5)(3,8)(6,7)(9,10)(11,14)(12,21)(13,18)(15,23)(16,22)(17,20)(19,24);(9,11,13,20)(10,14,18,17)(12,19,23,22)(15,16,21,24)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57393);