Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×#407 | C22 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C2×#407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,8,13,10,11,17,18,9,14,16,20,7,12,15,19);(1,2,3,4)(6,11)(7,10)(8,16)(12,14)(13,18)(19,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,g^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,g^b=gi,d^c=dh,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1278);