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 | |
2246 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)(11,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24)(25,26,30,31)(27,28,33,32)(29,37,40,34)(35,39,38,36);(5,6,7,8)(10,12)(15,17)(16,20)(23,24)(26,31)(27,33)(34,37)(35,38);(5,6)(7,8)(9,11,14,19)(10,15,12,17)(13,21,22,18)(16,24,20,23)(25,27,30,33)(26,32,31,28)(29,38,40,35)(34,36,37,39);(1,2)(3,4)(9,12,14,10)(11,17,19,15)(13,16,22,20)(18,23,21,24)(25,28,30,32)(26,33,31,27)(29,39,40,36)(34,35,37,38);(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,29)(26,34)(27,35)(28,36)(30,40)(31,37)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=f,e^2=fg,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, 3242597);