Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125890 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7750 |
| commutator, Frattini, cosocle | C43 | C23 |
| prime radical | #31076 | C2 |
| LCS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(1,2,6,9)(3,7,14,23)(4,15)(5,19,20,10)(8,17)(11,27)(12,22,13,26)(16,24)(18,25)(21,30)(28,31)(29,32);(1,3)(2,7)(4,11)(5,12)(6,14)(8,21)(9,23)(10,26)(13,20)(15,31)(16,30)(17,29)(18,28)(19,22)(24,27)(25,32);(1,4)(2,8)(3,11)(5,16)(6,17)(7,21)(9,15)(10,24)(12,30)(13,28)(14,29)(18,20)(19,25)(22,32)(23,31)(26,27)Polycyclic presentation:
2^9|a^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,d^b=df,e^b=eg,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33087);