Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | D4 |
| commutator | C4 | C4×C32 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle, prime radical | C23 | C4⋊C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
172 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,15,19,16,20,23,29,17,21,24,30,26,32,35,39,18,22,25,31,27,33,36,40,28,34,37,41,38,42,43,44);(2,4)(5,6,8,11,9,12,7,10)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1211);