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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2188 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)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,30,31)(27,34,37,32)(28,29,39,33)(35,40,38,36);(5,6,7,8)(13,21)(16,20)(18,24)(22,23)(26,31)(29,33)(32,34)(36,40);(5,6)(7,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,32)(28,38)(29,40)(30,37)(31,34)(33,36)(35,39);(1,2)(3,4)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28,30,39)(26,33,31,29)(27,35,37,38)(32,40,34,36);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,33)(26,28,31,39)(27,36,37,40)(32,35,34,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,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, 4040925);