Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7481 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1390 |
| commutator | C2×Q8 | C4×C8 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C2 | #7187 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,5,11,7,12,28,41,8,13,29,43,6,30,27,42)(3,14,16,44,18,45,51,64,19,46,52,60,17,53,25,57)(4,20,21,54,23,10,36,37,24,38,26,58,22,39,56,63)(9,31,32,15,33,47,40,49,35,55,61,50,34,59,62,48);(1,3,6,17,8,19,7,18)(2,9,12,33,13,35,30,34)(4,15,22,48,24,50,23,49)(5,25,27,52,29,51,28,16)(10,14,38,45,39,46,20,53)(11,40,41,61,43,62,42,32)(21,55,56,47,26,31,36,59)(37,57,58,44,63,64,54,60)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=df,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1289);