Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145011 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2041 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28458 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,5)(4,6)(7,8)(10,15)(11,19)(12,21)(13,20)(16,18)(17,23)(22,24);(2,5)(6,7)(10,16)(11,19)(12,21)(13,20)(15,22)(17,24)(18,23);(2,6)(4,8)(5,7)(9,10,13,22,14,17,20,18)(11,16,19,23,21,24,12,15)Polycyclic presentation:
2^9|c^2=dg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52227);