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 | #34593 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,26,23,27,22,32)(19,28,21,20,29,25,31,30);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,25,27,28)(20,26,30,32)(21,22,31,24);(10,14)(11,15)(17,20,24,25,23,30,22,28)(18,19,26,21,27,29,32,31);(2,5)(6,8)(10,14)(13,16)(17,21)(18,20)(19,22)(23,31)(24,29)(25,26)(27,30)(28,32);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,26,27,32)(19,21,29,31)(20,28,30,25)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dfi,e^b=ei,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319187);