Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1676 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)(3,4)(5,6)(7,8)(10,11)(12,16)(17,18,21,19)(20,23,24,22)(25,26)(27,31)(28,30)(29,32)(33,34)(35,39)(36,38)(37,40);(1,2,5,6)(3,4,7,8)(9,10,13,11)(12,14,16,15)(26,28)(27,33)(32,36)(35,37);(2,6)(4,8)(9,10,13,11)(12,15,16,14)(17,19,21,18)(20,23,24,22)(25,27,30,33)(26,31,28,34)(29,37,38,35)(32,40,36,39);(1,3,5,7)(2,4,6,8)(9,11,13,10)(12,15,16,14)(17,20)(18,22)(19,23)(21,24)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2135253);