Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1586 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)(3,5)(4,6)(7,8)(11,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26,30,31)(27,34,32,28)(29,37,40,33)(35,36,38,39);(1,2,4,6)(3,5,7,8)(10,13)(14,16)(26,31)(27,32)(33,37)(35,38);(3,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22)(25,27,30,32)(26,28,31,34)(29,38,40,35)(33,39,37,36);(1,2)(3,8)(4,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28)(26,32)(27,31)(29,39)(30,34)(33,35)(36,40)(37,38);(1,3)(2,5)(4,7)(6,8)(25,29)(26,33)(27,35)(28,36)(30,40)(31,37)(32,38)(34,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=i,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454251);