Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C82 | D4 |
| commutator | C4 | C8×C16 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1273 |
| prime radical | C23 | #322 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C82 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
210 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 126 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:
(1,2,4,9,5,8,12,15,6,10,13,16,3,7,11,14);(2,7,10,8)(9,14,16,15)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=ch,e^a=eh224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93);