Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146231 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2094 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,10,15,16,24,11,12,17,14,20,22,21,13,19,23,18);(2,6)(3,4)(7,8)(9,11,13,20,14,19,24,10)(12,17,15,21,23,18,22,16);(2,7)(3,4)(6,8)(10,16)(11,18)(12,22)(13,24)(15,23)(17,19)(20,21)Polycyclic presentation:
2^9|a^2=dg,b^2=de,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fhi,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, 53427);