Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,13,16)(11,14,12,15)(19,23)(22,24)(25,26)(27,33)(28,35)(29,37)(30,31)(32,38)(34,36)(39,40);(1,3,6,4)(2,5,7,8)(18,21)(22,24)(26,31)(28,33)(29,38)(34,39);(2,7)(3,4)(9,11,13,12)(10,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,39)(30,35)(31,33)(32,40)(34,38)(36,37);(1,4,6,3)(2,5,7,8)(9,10)(11,14)(12,15)(13,16)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,5)(2,3)(4,7)(6,8)(9,12,13,11)(10,15,16,14)(17,19,20,23)(18,22,21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,37)(33,40)(35,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=gi,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5365046);