Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | C2×#1326 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,7,12,27,42,8,13,28,43,6,29,31,41)(3,14,17,45,19,47,51,64,15,44,48,61,18,46,25,57)(4,20,22,53,23,10,36,37,24,38,26,58,30,39,56,63)(9,32,34,16,35,49,40,50,33,59,60,21,54,55,62,52);(1,3,6,18,8,15,7,19)(2,9,12,35,13,33,29,54)(4,21,23,52,24,16,30,50)(5,25,27,17,28,51,31,48)(10,14,20,46,39,44,38,47)(11,40,41,34,43,62,42,60)(22,32,56,55,26,59,36,49)(37,57,58,45,63,64,53,61);(3,15)(9,33)(14,44)(16,21)(17,48)(18,19)(25,51)(32,59)(34,60)(35,54)(40,62)(45,61)(46,47)(49,55)(50,52)(57,64)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dh,e^b=efi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56846);