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 | |
1544 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,21)(22,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,5,8)(2,4,6,7)(9,10,12,14)(11,15,16,13)(18,20)(19,24)(26,31)(29,38)(32,39)(36,40);(2,6)(4,7)(9,10,12,14)(11,13,16,15)(17,18,21,20)(19,23,24,22)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,23,24,22)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fhi,d^2=fgh,e^2=h,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, 1520655);