Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C2×AGL(1,ℤ/8) | C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14)(17,18,21,24,22,25,29,31,20,19,23,26,28,27,30,32);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,21,26,22,27,29,32,20,18,23,24,28,25,30,31);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(18,19)(24,26)(25,27)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45057);