Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12187 |
| commutator, Frattini, cosocle | #381 | C23 |
| prime radical | #12094 | C2 |
| LCS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(2,7)(3,11)(5,16)(8,22)(9,26)(10,21)(12,30)(13,17)(14,18)(19,24)(20,27)(23,28)(25,32)(29,31);(1,2)(3,8)(4,7)(5,9)(6,10)(11,28)(12,19)(13,25)(14,26)(15,21)(16,27)(17,24)(18,20)(22,30)(23,31)(29,32);(1,3)(2,8)(4,12,15,31)(5,17)(6,13)(7,19,21,23)(9,24)(10,25)(11,16)(14,29,18,30)(20,22,26,32)(27,28)Polycyclic presentation:
2^9|c^2=fgh,d^2=f,e^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,e^d=ehi,g^d=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72237836, #72237856, #72237866, … |
| maximal normal subgroup of | #72237836, #72237856, #72237866, … |
| maximal quotient of | #72238057, #72238059, #72238062 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58540);