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 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,15,12,14)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,5,4)(3,6,8,7)(11,12)(14,15)(19,24)(21,23)(26,30)(27,34)(28,37)(38,39);(9,11,13,12)(10,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,3,5,8)(2,6,4,7)(9,12,13,11)(10,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,d^2=fghi,e^2=fgi,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, 1719284);