Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1939 |
| prime radical | C23 | C8×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
213 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,10)(4,5)(8,15)(9,13)(11,12)(17,18,21,25,22,19,26,28,20,24,29,32,30,27,23,31);(1,2)(3,8)(4,13)(5,7)(6,9)(10,14)(11,16)(12,15)(17,19,21,28,22,24,26,32,20,27,29,31,30,18,23,25);(1,3,5,10,6,12,4,11)(2,8,7,14,9,15,13,16)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=e,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47605);