Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #124 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13264 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 211 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:
(1,2,6,13,7,14,28,42,8,12,29,34,31,41,55,58,9,15,30,43,32,44,56,62,5,22,17,45,27,52,47,63)(3,16,20,25,21,46,51,26,19,35,49,54,50,59,64,23,18,10,33,36,48,37,57,24,11,38,39,53,40,60,61,4);(1,3,17,39,32,48,55,64,8,19,6,20,27,40,56,57,9,18,29,49,7,21,47,61,5,11,30,33,31,50,28,51)(2,10,34,53,44,46,63,23,12,38,43,25,52,59,42,24,15,16,45,54,14,37,58,4,22,35,13,36,41,60,62,26);(1,4,5,24,9,23,8,26)(2,11,12,3,15,19,22,18)(6,16,17,38,30,10,29,35)(7,25,27,53,32,36,31,54)(13,39,34,20,43,49,45,33)(14,40,41,21,44,50,52,48)(28,46,47,60,56,37,55,59)(42,61,58,51,62,64,63,57)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=e,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,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72217183, #72225000, #72225008, … |
| maximal normal subgroup of | #72217183, #72225000, #72225008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60200);