Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,5)(4,7)(11,15)(14,16)(19,23)(21,24)(25,26,30,33)(27,28,35,31)(29,37,40,32)(34,39,38,36);(1,2)(3,4)(5,7)(6,8)(10,13)(14,16)(25,27,30,35)(26,31,33,28)(29,34,40,38)(32,39,37,36);(1,3,6,5)(2,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(25,27)(26,28)(29,38)(30,35)(31,33)(32,39)(34,40)(36,37);(1,4,6,7)(2,3,8,5)(17,19)(18,21)(20,23)(22,24)(25,28)(26,27)(29,39)(30,31)(32,38)(33,35)(34,37)(36,40);(1,5,6,3)(2,7,8,4)(9,11,12,15)(10,14,13,16)(25,29,30,40)(26,32,33,37)(27,34,35,38)(28,36,31,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4942224);