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 | |
1458 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)(3,7,8,6)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(3,8)(6,7)(10,13)(14,16)(17,19,21,18)(20,23,24,22)(26,29)(27,34)(32,38)(35,36);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,37)(38,39);(1,2)(3,6)(4,5)(7,8)(17,20)(18,22)(19,23)(21,24)(25,28,30,37)(26,32,29,38)(27,35,34,36)(31,39,33,40);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(25,29,30,26)(27,31,34,33)(28,32,37,38)(35,40,36,39)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,d^2=f,e^2=fgh,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, 4107219);