Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14,16,18,17,19,20,15);(5,7,8,12,9,6,10,11)(13,15,16,14)(17,18,20,19);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,f^b=fg,g^b=gi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47827);