Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C4×#97 | C22 |
| socle | C22 | #1288 |
| prime radical | C23 | #339 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C4×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
213 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,21,14)(12,17)(13,18)(15,22,23,24)(16,20);(10,14,16,23,18,19,17,15)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=g,f^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,d^b=dg,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 513);