Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1454 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,12,15)(17,18,22,20)(19,23,21,24)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,37)(38,39);(1,2)(3,7)(4,6)(5,8)(9,11,13,12)(10,14,16,15)(17,19,22,21)(18,23,20,24)(26,28)(29,37)(31,33)(34,38);(1,2,4,6)(3,5,8,7)(9,11,13,12)(10,14,16,15)(17,20,22,18)(19,23,21,24)(25,27)(26,31)(28,33)(29,38)(30,35)(32,40)(34,37)(36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,21)(18,24,20,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,15,16,14)(17,21,22,19)(18,23,20,24)(25,29,30,37)(26,32,28,36)(27,34,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=gi,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5110718);