Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2153 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)(9,10,14,11)(12,17,19,15)(13,16,22,18)(20,24,21,23)(25,26,30,33)(27,34,31,28)(29,37,39,32)(35,36,40,38);(6,8)(10,11)(12,19)(16,18)(20,21)(25,27,30,31)(26,28,33,34)(29,35,39,40)(32,36,37,38);(5,6,7,8)(9,11,14,10)(12,15,19,17)(13,18,22,16)(20,23,21,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,2)(3,4)(9,12,14,19)(10,15,11,17)(13,21,22,20)(16,24,18,23)(25,28)(26,31)(27,33)(29,38)(30,34)(32,35)(36,39)(37,40);(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,29,30,39)(26,32,33,37)(27,35,31,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 2643810);