Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150453 |
| 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 | #391 | C23 |
| prime radical | #3492 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,8,3,4,17)(6,26,21,23,28,29,24,19)(7,13,25,15,20,14,22,9)(10,27,18,32,16,31,12,30);(1,3)(2,8)(4,19)(5,23)(6,12)(7,14)(9,21)(10,22)(11,30)(13,20)(15,24)(16,25)(17,32)(18,28)(26,31)(27,29);(1,4,8,5)(2,9,3,15)(6,24,28,21)(7,22,20,25)(10,29,16,26)(11,13,17,14)(12,23,18,19)(27,30,31,32)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,e^d=ehi,f^d=fi,f^e=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57938);