Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C4×C8 |
| Frattini, cosocle | #2931 | C22 |
| socle | C2 | #7204 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 64 points:
(1,2,5,12,8,14,29,45,9,15,30,46,7,31,6,13)(3,16,18,49,20,51,55,64,21,39,56,62,19,50,28,61)(4,22,24,41,26,11,38,40,27,42,60,48,32,43,25,58)(10,33,34,23,36,53,44,57,37,59,35,17,47,52,63,54);(1,3,6,18,7,20,30,55,9,21,29,56,8,19,5,28)(2,10,13,35,31,36,46,63,15,37,45,34,14,47,12,44)(4,23,25,52,32,57,60,33,27,17,38,53,26,54,24,59)(11,39,41,62,22,50,58,61,43,16,48,49,42,51,40,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=gh,f^2=i,g^2=i,b^a=bc,c^a=cf,f^a=fh,c^b=ci,d^b=df,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1346);