Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C82 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1282 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21)(12,26,27,17)(13,18)(14,20)(15,19)(16,22,33,44)(23,41)(24,43)(25,42)(28,50,52,36)(29,51,53,37)(30,39)(31,38)(32,40)(34,45,56,64)(35,46,57,65)(47,62)(48,61)(49,63)(54,68,69,59)(55,60)(58,66,70,72)(67,71);(9,11,13,23,14,24,30,47,15,25,31,48,32,49,55,67)(10,16,18,34,20,35,39,58,19,33,38,56,40,57,60,70)(12,22,28,45,29,46,54,66,27,44,52,64,53,65,69,72)(17,21,36,41,37,43,59,62,26,42,50,61,51,63,68,71)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=h,h^2=i,b^a=bc,c^a=cf,f^a=fi,d^b=df,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 482);