Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125647 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #31071 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 24 points:
(1,2,5,8)(3,6,4,7)(10,11)(12,23)(13,16)(14,24)(17,20)(18,22)(19,21);(2,6)(3,4)(7,8)(9,10,15,19)(11,16,21,23)(12,17,13,22)(14,20,24,18);(2,7)(3,4)(6,8)(9,11)(10,16)(12,18)(13,20)(14,17)(15,21)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=fi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=df,e^b=eg,f^b=fi,d^c=dg,e^c=eh,h^c=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33478);