Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,12,16,14)(17,18,21,22)(19,23,24,20)(26,30)(28,31);(9,11,13,16)(10,14,15,12)(18,22)(20,23)(25,26,29,30)(27,28,32,31);(10,15)(12,14)(17,19,21,24)(18,23,22,20)(25,27,29,32)(26,31,30,28);(1,3,5,8)(2,6,4,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fg,d^2=fhi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4286826);