Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2094 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #496 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 266 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:
(1,2,4,7,6,3,5,8)(10,15,17,18)(16,21,24,23);(1,2)(3,6)(4,8)(5,7)(10,16,17,24)(11,13)(12,20)(15,23,18,21)(19,22);(1,3,6,2)(4,7,5,8)(9,10)(11,18)(12,21)(13,15)(14,17)(16,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=d,b^2=deh,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53472);