Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #466 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9594 |
| prime radical | C23 | #443 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,7,14,8,12,6,32,9,16,24,13,5,15,31,42)(3,17,21,45,19,10,20,35,22,36,47,44,30,37,52,34)(4,23,27,40,28,54,26,11,29,39,56,38,25,55,58,41)(18,46,50,61,48,53,49,43,51,62,63,33,57,59,64,60);(1,3,7,21,8,19,6,20,9,22,24,47,5,30,31,52)(2,10,14,35,12,36,32,44,16,37,13,34,15,17,42,45)(4,18,27,50,28,48,26,49,29,51,56,63,25,57,58,64)(11,33,39,59,38,60,55,46,41,61,23,53,40,43,54,62);(1,4,24,56,8,28,31,58,9,29,7,27,5,25,6,26)(2,11,32,55,15,40,14,39,16,41,42,54,12,38,13,23)(3,18,47,63,19,48,52,64,22,51,21,50,30,57,20,49)(10,33,44,46,17,43,35,59,37,61,45,62,36,60,34,53)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=eg,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320359, #72320403, #72320408, … |
| maximal normal subgroup of | #72320359, #72320403, #72320408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47545);