Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,7)(10,13)(11,15)(25,26)(27,28)(29,37)(30,31)(32,33)(34,38)(35,39)(36,40);(10,13)(14,16)(17,18,21,20)(19,23,24,22)(26,31)(27,33)(29,38)(36,39);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,23,24,22)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|b^2=fghi,c^2=gh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1034655);