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 | |
1758 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)(18,22)(19,20)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(18,22)(23,24)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,3)(2,6)(4,8)(5,7)(17,19,21,20)(18,23,22,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=hi,d^2=h,e^2=fg,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, 1059280);