Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1522 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)(3,5,7,8)(9,10,13,16)(11,12,15,14)(17,18,21,22)(19,23,24,20)(25,26,31,34)(27,32,28,29)(30,33,38,40)(35,39,36,37);(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,24)(22,23)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(11,15)(12,14)(18,22)(20,23)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,23)(22,24)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,3,4,7)(2,5,6,8)(9,12,13,14)(10,15,16,11)(17,18)(19,20)(21,22)(23,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgh,c^2=fgh,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5512898);