Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| 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 | |
275 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 86 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)(33,34,39,43,38,45,55,62,37,44,46,47,54,42,60,61)(35,36,49,52,48,53,59,64,50,40,56,57,63,51,41,58);(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)(33,35,46,49,38,48,60,59,37,50,39,56,54,63,55,41)(34,40,43,52,45,51,62,64,44,36,47,57,42,53,61,58);(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)(33,36,37,40)(34,41,44,59)(35,47,50,43)(38,53,54,51)(39,52,46,57)(42,56,45,49)(48,61,63,62)(55,64,60,58)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,e^a=ei,d^b=di,e^b=egi,f^b=fi,d^c=dgi,e^c=eg,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45035);