Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| socle | C23 | #336 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 2 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15,16,13)(12,14)(17,18,21,25,23,27,32,28,22,26,31,19,20,24,30,29);(2,5,7,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,27,23,29,32,26,22,25,31,24,20,28,30,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efg,d^b=dg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 535);