Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2362 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)(12,16,20,21)(18,23,24,22)(25,26,29,33)(27,31,35,30)(28,36,38,32)(34,39,37,40);(6,8)(10,14)(12,20)(15,17)(18,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,2)(3,4)(9,10)(11,17)(12,21)(13,14)(15,19)(16,20)(18,22)(23,24)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2976428);