Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2054 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33811 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,6)(4,7)(5,8)(10,18)(11,21)(12,14)(13,24)(15,16)(19,20)(22,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,12)(13,23)(14,18)(15,21)(16,20)(17,22)(19,24);(2,6)(3,4)(7,8)(9,11,12,10,16,18,13,21,17,19,24,22,15,23,14,20)Polycyclic presentation:
2^9|c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,h^b=hi,e^c=egh,f^c=fgh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52704);