Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133825 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2131 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 2131 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23)(18,21,24,22);(2,5)(3,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,23)(19,21)(20,24);(10,15)(11,17)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=f,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72225870, #72225989, #72226091, … |
| maximal normal subgroup of | #72225870, #72225989, #72226091, … |
| maximal quotient of | #72237726, #72237733, #72283777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41232);