Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| 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×D4 | C4×C8 |
| Frattini, cosocle | #2931 | C22 |
| socle | C2 | #7204 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ 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:
(1,2,5,12,8,14,29,43,9,15,30,44,7,31,6,13)(3,16,18,46,21,49,28,59,22,50,19,47,20,48,55,64)(4,23,24,39,26,11,37,38,27,40,58,63,32,41,25,57)(10,33,34,17,35,51,61,52,36,60,62,53,45,54,42,56);(1,3,6,19,7,21,30,55,9,22,29,18,8,20,5,28)(2,10,13,34,31,35,44,61,15,36,43,62,14,45,12,42)(4,17,25,51,32,52,58,60,27,53,37,54,26,56,24,33)(11,16,39,47,23,49,57,64,41,50,63,46,40,48,38,59)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=gh,f^2=i,g^2=i,b^a=bc,c^a=cf,f^a=fh,d^b=dfi,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, 1345);