Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52139 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,26,22,27,21,31)(19,28,29,32,30,24,20,25);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,30)(18,24,27,28)(20,23,29,21)(25,31,32,26);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,20,22,29)(18,25,27,32)(19,21,30,23)(24,26,28,31);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,18,21,31,22,27,23,26)(19,28,20,25,30,24,29,32);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,27,31)(19,29,30,20)(24,32,28,25)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6330959);