Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,13)(11,15)(17,18,20,22)(19,21,23,24)(25,26)(27,30)(28,33)(29,34)(31,35)(32,38)(36,39)(37,40);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(19,23)(21,24)(26,27)(28,35)(32,34)(36,37);(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,33,35,31)(29,34,38,32)(36,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24)(25,28)(26,31)(27,33)(29,37)(30,35)(32,40)(34,39)(36,38);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,18)(19,21)(20,22)(23,24)(25,29,30,38)(26,32,27,34)(28,36,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=f,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2613485);