Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150198 |
| 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 | C24 | #99 |
| 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 | |
453 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 453 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,12,17,14,19,25,34)(11,20,22,36,21,31,37,33)(13,16,28,32,27,18,26,35)(15,24,30,38,29,39,40,23);(1,2)(3,7)(4,6)(5,8)(9,11,13,23,14,21,27,38)(10,15,18,31,19,29,16,20)(12,24,26,22,25,39,28,37)(17,33,32,30,34,36,35,40);(11,21)(15,29)(20,31)(22,37)(23,38)(24,39)(30,40)(33,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=dg,e^b=efghi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57472);