Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1488 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,12,16,14)(19,23)(22,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(2,4)(6,8)(9,11,13,16)(10,14,15,12)(18,21)(22,24)(29,38)(32,37)(34,40)(36,39);(2,4)(3,7)(10,15)(12,14)(17,18,20,21)(19,22,23,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,4,5,2)(3,6,7,8)(9,12,13,14)(10,11,15,16)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,c^2=fgh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4106071);