Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | #16810 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2033 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(2,3)(4,8)(9,11)(10,13)(12,14)(15,16)(18,21)(19,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(2,3)(6,7)(10,15)(11,12)(17,18,20,21)(19,22,24,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,15,13)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,16)(10,11,15,12)(17,19,20,24)(18,22,21,23)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=fg,c^2=fi,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431394);