Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #390 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13190 |
| prime radical | #525 | C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,6,14,7,15,33,49,8,13,34,47,36,48,56,62,9,16,35,50,37,51,63,60,5,12,18,39,31,19,40,54)(3,17,21,29,22,52,55,61,20,43,53,26,11,44,45,59,23,41,32,4,24,27,57,28,30,10,38,25,46,42,64,58);(1,3,18,53,37,24,56,64,8,20,6,32,31,46,63,55,9,23,34,38,7,22,40,45,5,30,35,21,36,11,33,57)(2,10,39,29,51,44,62,28,13,17,14,26,19,27,60,58,16,43,47,4,15,42,54,61,12,41,50,25,48,52,49,59);(1,4,5,26,9,29,8,25)(2,11,13,24,16,46,12,22)(3,19,20,15,23,48,30,51)(6,27,18,44,35,52,34,42)(7,28,31,59,37,61,36,58)(10,40,41,63,43,56,17,33)(14,45,47,57,50,64,39,55)(21,54,53,49,32,62,38,60)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=i,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72217183, #72224998, #72225006, … |
| maximal normal subgroup of | #72217183, #72224998, #72225006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60083);