Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18,22,24)(19,20,21,23)(26,28)(30,32);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,22,21)(18,23,24,20)(27,31)(30,32);(3,7)(6,8)(11,15)(14,16)(18,24)(19,21)(25,26,29,28)(27,30,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,32)(29,31);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,20,24,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2066825);