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 | |
1524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(11,15)(12,14)(17,18,21,20)(19,22,24,23)(25,26)(27,33)(28,35)(29,37)(30,31)(32,38)(34,36)(39,40);(3,8)(5,7)(9,10,13,16)(11,12,15,14)(17,19,21,24)(18,22,20,23)(26,31)(28,33)(29,38)(34,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,14,16,12)(17,19)(18,22)(20,23)(21,24)(25,27,30,35)(26,28,31,33)(29,34,38,40)(32,36,37,39);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,15)(13,16)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12,13,14)(10,15,16,11)(17,20,21,18)(19,23,24,22)(25,29)(26,32)(27,34)(28,36)(30,38)(31,37)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fgi,d^2=fi,e^2=h,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, 1658911);