Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | |
1602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10,14,12)(11,16,15,13)(17,18,21,24)(19,23,20,22)(25,26,30,31)(27,33,34,28)(29,37,39,32)(35,36,40,38);(1,3,5,7)(2,6,8,4)(17,18,21,24)(19,22,20,23)(26,31)(27,34)(32,37)(35,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27,30,34)(26,28,31,33)(29,35,39,40)(32,36,37,38);(1,2)(3,6)(4,7)(5,8)(9,12,14,10)(11,16,15,13)(17,19,21,20)(18,22,24,23)(25,28)(26,27)(29,38)(30,33)(31,34)(32,40)(35,37)(36,39);(1,4,5,6)(2,7,8,3)(9,13,14,16)(10,15,12,11)(17,20,21,19)(18,23,24,22)(25,29,30,39)(26,32,31,37)(27,35,34,40)(28,36,33,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1319137);