Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #2813 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1346 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)(10,13)(11,15)(17,18,22,26,21,24,30,32)(19,20,28,29,27,23,25,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(19,27)(20,23)(25,28)(29,31);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,24)(21,27)(22,28)(25,30)(26,31)(29,32);(2,6)(5,8)(10,13)(14,16)(18,24)(19,27)(25,28)(26,32);(17,20,22,29,21,23,30,31)(18,25,26,19,24,28,32,27)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfg,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315123);