Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140203 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1934 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9558 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,7,14,9,15,6,35,10,16,34,13,8,38,5,32)(3,17,20,48,22,49,33,11,23,40,36,41,21,55,19,42)(4,24,28,57,30,58,26,12,31,44,37,45,29,61,27,46)(18,43,51,56,25,59,50,62,53,63,54,64,52,39,60,47);(1,3,8,21,10,23,9,22)(2,11)(4,25,30,53,31,52,29,18)(5,33,34,20,6,19,7,36)(12,43,46,63)(13,40)(14,17)(15,41)(16,42)(24,56,44,64)(26,51,37,50,27,54,28,60)(32,55)(35,49)(38,48)(39,45,59,57)(47,58,62,61);(1,4,5,26,8,29,34,28,10,31,6,27,9,30,7,37)(2,12,13,44,15,45,32,61,16,46,14,24,38,57,35,58)(3,18,19,50,21,52,36,51,23,53,33,60,22,25,20,54)(11,39,17,47,41,43,49,56,42,59,40,62,48,63,55,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224946, #72224947, #72224958, … |
| maximal normal subgroup of | #72224946, #72224947, #72224958, … |
| maximal quotient of | C2×#140203 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47399);