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 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,13,14)(11,15,16,12)(17,18,21,19)(20,23,24,22)(25,26)(27,29)(28,35)(30,31)(32,37)(33,38)(34,40)(36,39);(1,2,6,4)(3,7,5,8)(10,14)(11,16)(17,19,21,18)(20,23,24,22)(26,31)(29,38)(32,35)(36,40);(1,3,6,5)(2,7,4,8)(9,11,13,16)(10,12,14,15)(25,27)(26,29)(28,34)(30,33)(31,38)(32,36)(35,40)(37,39);(1,4,6,2)(3,7,5,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,37)(26,32,31,35)(27,34,33,39)(29,36,38,40);(1,5,6,3)(2,7,4,8)(9,12)(10,11)(13,15)(14,16)(25,29)(26,33)(27,31)(28,36)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,c^2=g,d^2=fg,e^2=gh,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, 5397710);