Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,14,12,15)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(3,4)(6,7)(18,21)(22,24)(26,31)(27,34)(29,37)(36,39);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,18,20,21)(19,22,23,24)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4,5,3)(2,7,8,6)(9,10)(11,14)(12,15)(13,16)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(1,2)(3,6)(4,7)(5,8)(9,12,13,11)(10,15,16,14)(17,19,20,23)(18,22,21,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=fi,c^2=fghi,d^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1746485);