Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,14,15)(11,12,17,16)(13,21,24,18)(19,23,22,20)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(6,8)(10,15)(11,17)(18,21)(19,22)(25,27,30,35)(26,31,28,32)(29,34,38,37)(33,39,36,40);(5,6,7,8)(9,11,14,17)(10,16,15,12)(13,22,24,19)(18,20,21,23)(25,28,30,26)(27,31,35,32)(29,33,38,36)(34,40,37,39);(1,2)(3,4)(9,12,14,16)(10,17,15,11)(13,23,24,20)(18,19,21,22)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(9,13)(10,18)(11,19)(12,20)(14,24)(15,21)(16,23)(17,22)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3244332);