Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| 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 | |
2112 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)(3,4)(5,7)(6,8)(10,14)(11,16)(17,18,21,20)(19,22,24,23)(25,26)(27,29)(28,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,16,15)(18,20)(22,23)(25,27,30,31)(26,29,32,28);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,15,14,12)(17,18,21,20)(19,23,24,22)(25,27)(26,28)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,21,24)(18,22,20,23)(25,28,30,29)(26,31,32,27);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20,21,18)(19,23,24,22)(25,29,30,28)(26,27,32,31)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 603898);