Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C4⋊C4 |
| prime radical | C23 | C8×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(2,7,9,13)(8,14,15,16)(17,18,21,25,22,26,30,32,20,23,27,31,29,19,24,28);(1,2,5,7,6,9,4,13)(3,8,11,14,12,15,10,16)(18,23)(19,26)(25,31)(28,32);(1,3,4,10,6,12,5,11)(2,8,13,16,9,15,7,14)(17,19,27,32,22,18,24,31,20,26,21,28,29,23,30,25)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^b=ci,d^b=dg,e^b=eg176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36388);