Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #98566 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| prime radical | #59816 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
1360 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,14)(12,15)(13,16)(17,18,19,22,20,21,23,24);(2,6)(3,4)(7,8)(10,14)(11,12)(15,16);(10,15)(11,12)(14,16)(18,21)(19,23)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=g,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg,f^c=fgi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5886);