Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| 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×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2300 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)(9,10,12,14)(11,15,16,13)(17,18,22,21)(19,24,20,23)(25,26)(27,31)(28,30)(29,32);(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,20)(18,23,21,24)(25,27)(26,31)(28,29)(30,32);(2,3)(6,7)(11,16)(13,15)(18,21)(19,20)(26,28)(27,32);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,19)(18,24,21,23)(25,28,30,26)(27,29,32,31);(1,4)(2,6)(3,7)(5,8)(9,11,12,16)(10,13,14,15)(17,21,22,18)(19,24,20,23)(25,29)(26,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389393);