Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2194 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)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(26,31)(27,34)(29,37)(36,39);(6,8)(10,15)(11,17)(13,21)(19,24)(25,26,30,31)(27,28,34,32)(29,38,37,33)(35,39,40,36);(5,6,7,8)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,2)(3,4)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,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, 1414189);