Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,5)(6,7)(10,13)(11,15)(17,18)(19,22)(20,21)(23,24)(25,26,30,31)(27,29,33,39)(28,35,38,32)(34,40,36,37);(1,3)(2,5)(4,6)(7,8)(9,10,12,13)(11,16,15,14)(18,20)(19,23)(26,31)(29,39)(32,35)(37,40);(2,7)(3,4)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,24,23,22)(25,27,30,33)(26,29,31,39)(28,36,38,34)(32,40,35,37);(1,4,6,3)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24)(25,28,30,38)(26,32,31,35)(27,34,33,36)(29,37,39,40);(1,5,6,8)(2,3,7,4)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)(25,29)(26,33)(27,31)(28,37)(30,39)(32,36)(34,35)(38,40)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=hi,d^2=fgh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1520653);