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 | |
1720 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)(3,6,4,7)(9,10,14,16)(11,12,15,13)(18,21)(19,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,3,5,4)(2,6,8,7)(9,11,14,15)(10,12,16,13)(18,21)(22,24)(26,31)(27,34)(29,37)(36,39);(9,12,14,13)(10,11,16,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,2)(3,6)(4,7)(5,8)(9,13,14,12)(10,15,16,11)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(1,4,5,3)(2,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,19,20,23)(18,22,21,24)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=gh,d^2=gh,e^2=gi,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, 1161568);