Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1634 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)(9,10,14,16)(11,13,12,15)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,36)(29,30)(32,38)(33,35)(34,40)(37,39);(2,5)(3,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,24)(18,22,20,23)(26,29)(27,33)(32,36)(34,37);(2,5)(6,8)(10,16)(11,12)(18,20)(22,23)(25,27,30,33)(26,31,29,35)(28,37,38,34)(32,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,16,15)(17,20,21,18)(19,22,24,23)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,23)(20,22)(21,24)(25,29,30,26)(27,35,33,31)(28,36,38,32)(34,40,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fg,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4417010);