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 | |
1536 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)(10,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,3,5,8)(2,4,6,7)(11,15)(14,16)(18,22)(19,24)(27,34)(29,37)(31,33)(32,36);(2,6)(3,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,23,22,20)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,22)(21,24)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,21,23)(18,19,22,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fghi,c^2=fg,d^2=fh,e^2=fg,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, 4976377);