Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #512110 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #33004 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2873 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,15)(5,19)(8,24)(9,10)(11,17)(13,30)(14,31)(16,32)(18,21)(20,23)(22,26)(25,28);(4,16)(7,10)(12,30)(15,32)(17,23)(18,31)(22,28)(25,26);(1,2,5,20,6,9,19,11)(3,8,21,27,13,24,14,29)(4,17,15,10,16,23,32,7)(12,28,31,25,30,22,18,26);(1,3)(2,8)(4,18)(5,21)(6,13)(7,22)(9,24)(10,28)(11,29)(12,15)(14,19)(16,31)(17,26)(20,27)(23,25)(30,32)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,e^d=ei,f^d=fh,g^d=gh,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419306);