Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #322 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | #346 | C23 |
| socle | C2 | C2×#1313 |
| prime radical | C23 | #347 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
161 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,4,9,5,11,20,32,6,10,21,34,23,35,52,62,7,12,19,33,24,38,50,61,25,36,44,31,55,39,64,59)(3,13,15,40,16,29,45,57,17,41,46,60,47,27,51,56,14,37,42,63,22,53,54,30,43,58,18,48,49,8,26,28);(1,3)(2,8,10,29,12,27,36,53)(4,18,19,46)(5,22)(6,17)(7,14)(9,30,31,56,33,57,34,28)(11,37,35,58,38,13,39,41)(15,44,42,21)(16,24)(20,51,50,26)(23,49)(25,43)(32,60,59,40,61,48,62,63)(45,52,54,64)(47,55);(3,14)(8,27)(13,37)(15,42)(16,22)(17,43)(18,46)(26,51)(28,56)(29,53)(30,57)(40,63)(41,58)(45,54)(47,49)(48,60)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bf,e^a=ehi,g^a=gi,c^b=ci,d^b=dhi,f^b=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72217187, #72217188, #72217199, … |
| maximal normal subgroup of | #72217187, #72217188, #72217199, … |
| maximal quotient of | #72223655, C2×#149504, #72231870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56700);