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 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,4,5,7)(11,15)(14,16)(19,20)(23,24)(25,26,29,30)(27,31,33,37)(28,32,35,36)(34,38,39,40);(1,3,6,5)(2,7,8,4)(10,13)(14,16)(18,22)(23,24)(26,30)(27,33)(28,35)(38,40);(3,5)(4,7)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,36)(29,30)(32,35)(33,37)(34,40)(38,39);(1,4,6,7)(2,3,8,5)(17,19,21,20)(18,23,22,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,39)(36,40);(1,5,6,3)(2,7,8,4)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,28)(26,32)(27,34)(29,35)(30,36)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,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, 5238302);