Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,23,24,21)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(3,8)(5,7)(9,11,13,16)(10,14,15,12)(26,31)(28,32)(29,37)(34,40);(10,15)(11,16)(17,19,20,24)(18,21,22,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5900376);