Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,23,24,22)(25,26)(27,34)(28,30)(29,32)(31,33)(35,37)(36,39)(38,40);(2,5)(3,7)(10,13)(11,15)(17,18)(19,22)(20,21)(23,24)(25,27,31,28)(26,30,33,34)(29,35,38,36)(32,39,40,37);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,28,31,27)(26,30,33,34)(29,35,38,36)(32,37,40,39);(1,2)(3,8)(4,5)(6,7)(9,10)(11,16)(12,13)(14,15)(17,19)(18,23)(20,22)(21,24)(25,29,31,38)(26,32,33,40)(27,35,28,36)(30,39,34,37);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,30)(26,27)(28,33)(29,37)(31,34)(32,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fi,d^2=fg,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3143511);