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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2210 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)(9,10,14,17)(11,18,15,12)(13,21,22,16)(19,20,23,24)(25,26)(27,34)(28,32)(29,31)(30,37)(33,38)(35,39)(36,40);(6,8)(11,15)(12,18)(19,23)(20,24)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(5,6)(7,8)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,2)(3,4)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,29,31,26)(27,34,28,32)(30,33,38,37)(35,39,36,40);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=f,e^2=f,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, 2976423);