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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(5,6,7,8)(12,21)(15,20)(18,24)(22,23)(26,29)(31,33)(32,36)(39,40);(5,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27,30,35)(26,31,29,33)(28,37,38,34)(32,40,36,39);(1,2)(3,4)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28,30,38)(26,32,29,36)(27,34,35,37)(31,39,33,40);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,26)(27,31,35,33)(28,32,38,36)(34,40,37,39)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=f,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, 4040674);