Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1548 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)(3,4,8,7)(9,10,12,14)(11,13,15,16)(19,23)(22,24)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,37)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(18,21)(22,24)(26,28)(29,37)(31,33)(34,38);(2,6)(3,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,33)(29,38)(30,35)(32,40)(34,37)(36,39);(1,3)(2,7)(4,6)(5,8)(17,19)(18,22)(20,23)(21,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,4,5,7)(2,3,6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,32,28,36)(27,34,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4972533);