Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(10,14)(12,19)(15,20)(18,23)(25,26)(27,30)(28,31)(29,34)(32,38)(33,35)(36,40)(37,39);(5,6,7,8)(10,14)(11,17)(16,21)(18,23)(26,27)(28,35)(32,34)(36,37);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,33,35,31)(29,34,38,32)(36,40,37,39);(1,2)(3,4)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,35)(26,31,27,33)(29,37,38,36)(32,40,34,39);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2514565);