Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149895 |
| 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 | C22×D4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | #3484 | C4 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 522 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:
(2,7,9,22,10,24,17,32)(3,12,13,29)(4,16)(8,21,26,27,11,25,28,23)(14,30,15,18)(19,31);(1,2)(3,7)(4,17)(5,8)(6,10)(9,19)(11,20)(12,27)(13,25)(14,21)(15,24)(16,28)(18,23)(22,30)(26,31)(29,32);(2,8)(7,21)(9,26)(10,11)(17,28)(22,27)(23,32)(24,25)Polycyclic presentation:
2^9|a^2=e,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cgh,e^b=efi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57091);