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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(11,15)(14,16)(17,18,21,20)(19,22,24,23)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(2,6)(4,7)(10,13)(14,16)(17,18,21,20)(19,23,24,22)(29,38)(32,37)(34,40)(36,39);(1,3,5,8)(2,4,6,7)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3)(2,7)(4,6)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,4,5,7)(2,8,6,3)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=gh,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, 4106068);