Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | |
1716 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)(3,7,4,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,19)(20,24,21,23)(25,26)(27,30)(28,33)(29,34)(31,36)(32,38)(35,39)(37,40);(9,11,14,13)(10,15,12,16)(17,19,22,18)(20,24,21,23)(26,27)(31,33)(32,34)(39,40);(1,3,5,4)(2,6,8,7)(9,12,14,10)(11,15,13,16)(17,20,22,21)(18,23,19,24)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,2)(3,6)(4,7)(5,8)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,4,5,3)(2,7,8,6)(9,13,14,11)(10,15,12,16)(17,21,22,20)(18,24,19,23)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1306369);