Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150578 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | #3460 | 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 | |
381 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 54 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)(10,15,17,24,18,20,16,23)(12,19,21,22);(1,2)(3,6)(4,8)(5,7)(9,10)(11,16)(12,20)(13,17)(14,18)(15,21)(19,24)(22,23);(1,3,5,4)(2,6,7,8)(9,11,14,13)(10,16,18,17)(12,19,21,22)(15,23,20,24)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cgi,d^a=df,f^a=fh,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57909);