Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #99 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | #11465 |
| prime radical | #525 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,4,10,5,11,18,30,6,12,19,31,20,32,41,52,7,8,17,24,21,25,39,46,22,26,40,47,42,48,57,61)(3,9,13,27,14,28,34,49,15,29,35,50,36,51,55,62,16,23,33,43,37,44,53,58,38,45,54,59,56,60,63,64);(2,8)(9,23)(11,25)(12,26)(28,44)(29,45)(32,48)(51,60);(1,3,5,14,6,15,20,36,7,16,21,37,22,38,42,56)(2,9,11,28,12,29,32,51,8,23,25,44,26,45,48,60)(4,13,18,34,19,35,41,55,17,33,39,53,40,54,57,63)(10,27,30,49,31,50,52,62,24,43,46,58,47,59,61,64)Polycyclic presentation:
2^9|a^2=e,c^2=g,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,e^b=ef,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287376, #72287798, #72287807, … |
| maximal normal subgroup of | #72287376, #72287798, #72287807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56865);