Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144637 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10390 |
| commutator, Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9)(3,11,13,7)(5,10,17,23)(6,8,16,21)(12,26,27,19)(14,24,29,31)(15,25,28,20)(18,22,30,32);(1,3,4,13)(2,7,9,11)(5,12,17,27)(6,15,16,28)(8,20,21,25)(10,19,23,26)(14,30,29,18)(22,31,32,24);(2,8)(3,12)(5,18)(7,19)(11,24)(13,27)(14,15)(17,30)(20,31)(23,32)(25,26)(28,29)Polycyclic presentation:
2^9|a^2=d,b^2=d,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,f^c=fi,e^d=ei29 conjugacy classes — view the full character table.
| maximal subgroup of | #72235133 |
| maximal normal subgroup of | #72235133 |
| maximal quotient of | #72235170, #72235245, #72235249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51833);