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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1632 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)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(2,5)(3,7)(9,11,13,16)(10,12,14,15)(17,18,20,21)(19,23,24,22)(26,30)(27,33)(32,36)(35,39);(2,5)(6,8)(10,14)(12,15)(18,21)(22,23)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19,20,24)(18,22,21,23)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=h,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, 5145451);