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 | |
1713 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)(9,10,12,14)(11,15,16,13)(18,22)(20,24)(26,30)(27,33)(32,36)(35,37);(3,4)(6,7)(11,16)(13,15)(18,22)(19,23)(25,26,29,30)(27,34,33,31)(28,36,38,32)(35,39,37,40);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,4,5,3)(2,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,38)(36,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fghi,d^2=fghi,e^2=f,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, 3098760);