Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150405 |
| 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 | #382 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(2,6)(3,4)(7,8)(10,15,16,23,18,22,17,24)(12,19,20,21);(1,2,5,8)(3,6,4,7)(9,10,13,16,14,18,11,17)(12,15,19,24,20,22,21,23);(1,3,5,4)(2,7,8,6)(9,11,14,13)(12,19,20,21)(15,22)(23,24)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=cgi,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efghi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^d=ehi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58126);