Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123275 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C82 | C23 |
| prime radical | #12424 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
258 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,9)(4,11)(7,16)(8,15)(10,13)(12,14)(18,22)(19,26)(20,28)(23,27)(24,31)(25,30)(29,32);(1,2,5,8)(3,10,12,16)(4,13,11,7)(6,14,15,9)(19,26)(20,28)(22,27)(23,25)(29,32)(30,31);(17,18)(19,27)(20,23)(21,24)(22,26)(25,28)(29,30)(31,32)Polycyclic presentation:
2^9|b^2=h,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,d^b=df,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225213, #72226835, #72226921, … |
| maximal normal subgroup of | #72225213, #72226835, #72226921, … |
| maximal quotient of | #72225188, #72225193 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30474);