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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(4,6)(17,18,21,19)(20,23,24,22)(25,26,30,29)(27,31,35,34)(28,36,38,32)(33,40,37,39);(1,2,5,8)(3,4,7,6)(9,10,12,14)(11,13,15,16)(17,19,21,18)(20,23,24,22)(26,29)(31,34)(32,36)(39,40);(1,3,5,7)(2,6,8,4)(9,10)(11,13)(12,14)(15,16)(25,27,30,35)(26,31,29,34)(28,37,38,33)(32,40,36,39);(1,2)(3,6)(4,7)(5,8)(17,20)(18,22)(19,23)(21,24)(25,28,30,38)(26,32,29,36)(27,33,35,37)(31,39,34,40);(1,4,5,6)(2,7,8,3)(9,11,12,15)(10,13,14,16)(25,29,30,26)(27,34,35,31)(28,36,38,32)(33,40,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=fg,d^2=f,e^2=fh,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, 1731086);