Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #384 | C23 |
| prime radical | #3492 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(2,5,6,7)(9,10)(11,18,20,23,21,24,19,14)(12,16)(13,15)(17,22);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,19)(13,21)(15,23)(16,18)(17,24)(20,22);(10,15)(11,19,21,20)(14,18,23,24)(16,17)Polycyclic presentation:
2^9|a^2=e,c^2=hi,d^2=g,e^2=hi,f^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=dhi,e^d=ehi,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57656);