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 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,14)(11,15,16,12)(19,20)(23,24)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(2,6)(3,7)(10,14)(11,16)(18,22)(23,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(3,7)(4,8)(9,11,13,16)(10,15,14,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,20)(18,23,22,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,4)(2,3)(5,8)(6,7)(9,12,13,15)(10,11,14,16)(17,20,21,19)(18,24,22,23)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ghi,d^2=i,e^2=gh,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, 5238300);