Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C23 | #435 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,12)(4,16,19,26)(5,20)(6,10)(7,11)(8,14)(9,18,25,17)(13,28,30,22)(15,23)(21,24)(27,29,32,31)(33,34,37,42,38,43,52,60,39,44,53,61,36,41,46,55)(35,45,48,62,47,57,63,59,49,58,51,64,50,40,54,56);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,18,31,19,30,17,29)(9,22,26,27,25,28,16,32)(33,35,46,63,36,50,53,48,39,49,52,54,38,47,37,51)(34,40,55,62,41,58,61,56,44,57,60,64,43,45,42,59);(1,4,7,19)(2,9,11,25)(3,13,15,30)(5,18,6,17)(8,22,24,28)(10,26,20,16)(12,27,23,32)(14,31,21,29)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,e^b=eg,f^b=fi,d^c=dg,e^c=eg,f^c=fg86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45032);