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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2405 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)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,23,22,24)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(5,6,7,8)(10,14)(12,21)(15,17)(18,22)(27,34)(29,37)(31,33)(32,36);(5,6)(7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,2)(3,4)(9,11,13,19)(10,15,14,17)(12,22,21,18)(16,24,20,23)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(9,12,13,21)(10,16,14,20)(11,18,19,22)(15,23,17,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fg,d^2=g,e^2=g,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, 4040989);