Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125652 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7734 |
| 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 | |
480 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 480 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:
(1,2,6,10)(3,7,14,23)(4,8,16,25)(5,18,19,9)(11,27,29,32)(12,22,13,21)(15,26,17,24)(20,30,31,28);(1,3)(2,7)(4,11)(5,12)(6,14)(8,20)(9,21)(10,23)(13,19)(15,30)(16,29)(17,28)(18,22)(24,32)(25,31)(26,27);(1,4)(2,8)(3,11)(5,15)(6,16)(7,20)(9,24)(10,25)(12,30)(13,28)(14,29)(17,19)(18,26)(21,32)(22,27)(23,31)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=h,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=eh,f^c=fi,g^c=gi41 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267788 |
| maximal subgroup of | #72267788 |
| maximal normal subgroup of | #72267788 |
| maximal quotient of | #72267790 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32849);