Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(12,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,27,30,35)(26,31,33,28)(29,37,38,34)(32,36,40,39);(10,11)(14,15)(18,22)(19,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,20,22,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,22)(21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5267054);