Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,14)(11,16)(17,18,22,23)(19,24,21,20)(25,26,30,32)(27,33,35,28)(29,37,38,31)(34,36,40,39);(1,3,6,5)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,21)(18,20,23,24)(29,38)(31,37)(34,40)(36,39);(3,5)(7,8)(10,14)(12,15)(17,20,22,24)(18,19,23,21)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,4,6,2)(3,8,5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,16,14,11)(17,21,22,19)(18,20,23,24)(25,29)(26,31)(27,34)(28,36)(30,38)(32,37)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=fgh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4151909);