Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10,13,12)(11,14,15,16)(17,18,20,22)(19,21,23,24)(25,26)(27,33)(28,35)(29,37)(30,31)(32,38)(34,36)(39,40);(3,7)(4,6)(11,15)(14,16)(17,19,20,23)(18,21,22,24)(26,31)(28,33)(29,38)(34,40);(1,3,5,7)(2,6,8,4)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,23)(22,24)(25,27,30,35)(26,28,31,33)(29,34,38,40)(32,36,37,39);(1,4,5,6)(2,7,8,3)(9,12,13,10)(11,16,15,14)(17,18)(19,21)(20,22)(23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,2)(3,6)(4,7)(5,8)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1556374);