Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116849 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1924 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(2,6)(3,8)(4,7)(9,10,13,19,15,11,16,21,14,17,20,24,12,18,22,23);(1,2,7,8,5,3,4,6)(9,11,20,21,15,17,22,24,14,18,13,23,12,10,16,19);(1,3,4,8,5,2,7,6)(10,11,17,18)(13,20)(16,22)(19,23,24,21)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=cgh,d^b=di,f^b=fi,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47404);