Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×C4⋊C16 |
| prime radical | C23 | C4⋊C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
152 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 152 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) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,13,19,15,21,29,36,16,18,30,33,25,34,38,40,14,20,28,35,23,17,31,32,12,26,27,39,11,22,24,37);(1,2,3,5,4,6,7,8)(9,11,12,23,14,25,16,15)(10,17,18,22,20,21,26,34)(13,24,27,31,28,38,30,29)(19,32,33,37,35,36,39,40);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=e,b^2=d,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,c^b=cfi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58243);