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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1928 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)(10,11)(14,15)(17,18)(19,20)(21,24)(22,23)(26,30)(27,33)(32,37)(35,39);(2,6)(3,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,20,23,22)(28,36)(32,37)(35,39)(38,40);(3,8)(4,7)(10,11)(12,16)(19,23)(20,22)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,24,20)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,4,5,7)(2,8,6,3)(9,12,13,16)(10,14,11,15)(17,20,21,22)(18,23,24,19)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=ghi,d^2=f,e^2=f,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, 5143944);