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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1356 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,12,15)(17,18,22,20)(19,23,21,24)(25,26,29,30)(27,31,35,33)(28,36,38,32)(34,40,37,39);(1,3,6,4)(2,7,8,5)(17,19,22,21)(18,23,20,24)(26,30)(31,33)(32,36)(39,40);(2,8)(3,4)(9,11,13,12)(10,14,16,15)(17,20,22,18)(19,23,21,24)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,4,6,3)(2,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,19,22,21)(18,24,20,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,5,6,7)(2,3,8,4)(9,12,13,11)(10,15,16,14)(17,21,22,19)(18,23,20,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=fgi,d^2=h,e^2=fi,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, 5358607);