Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | #2931 | C22 |
| socle | C2 | #7205 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 201 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:
(1,2,4,10,6,9,23,32,7,13,20,36,27,12,22,40)(3,14,16,41,18,8,19,28,15,30,43,34,45,31,48,57)(5,11,24,37,25,33,46,55,26,39,21,35,52,38,53,61)(17,42,49,62,50,29,54,58,44,59,47,63,64,60,51,56);(1,3,5,17,6,18,25,50,7,15,26,44,27,45,52,64)(2,8,11,29,9,30,33,59,13,31,39,60,12,14,38,42)(4,19,21,51,23,43,53,49,20,48,24,54,22,16,46,47)(10,34,35,62,32,57,61,58,36,41,37,63,40,28,55,56)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,h^a=hi,c^b=ci,d^b=dfhi,d^c=dhi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1359);