Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 310 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:
(2,6)(3,7)(4,8)(9,10,13,17,14,19,11,15)(12,21,23,24,22,16,20,18);(1,2,5,3)(4,7,8,6)(10,15,19,17)(12,22)(16,18,21,24)(20,23);(1,3,5,2)(4,6,8,7)(10,16)(11,13)(12,23)(15,18)(17,24)(19,21)(20,22)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=gi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dgi,e^b=eg,d^c=dg,e^c=ehi,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72342430, #72342436, #72342441, … |
| maximal normal subgroup of | #72342430, #72342436, #72342441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43743);