Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 128 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,26,28,25)(14,24)(16,23,32,30)(19,22)(21,27,29,31)(33,34,36,41,38,42,49,56,37,40,48,53,50,54,61,63)(35,43,45,55,46,57,59,64,44,39,47,51,58,52,60,62);(1,3,6,14)(2,7,10,22)(4,12,17,28)(5,19,20,13)(8,21,15,29)(9,24,18,11)(16,31,32,27)(23,26,30,25)(33,35,37,44)(34,39,40,43)(36,47,48,45)(38,46,50,58)(41,55,53,51)(42,52,54,57)(49,60,61,59)(56,64,63,62);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,23)(10,15)(11,25)(13,27)(14,28)(18,30)(19,31)(20,32)(22,29)(24,26)(35,44)(39,43)(45,47)(46,58)(51,55)(52,57)(59,60)(62,64)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,d^b=dgi,f^b=fi,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36889);