Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,30)(27,31,35,33)(28,36,38,32)(34,40,37,39);(9,11,14,12)(10,15,16,13)(17,19,21,24)(18,22,20,23)(26,30)(31,33)(32,36)(39,40);(10,16)(11,12)(18,20)(22,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,16,13)(17,20,21,18)(19,23,24,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,11,16,12)(17,19)(18,22)(20,23)(21,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5267116);