Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,33)(28,32)(29,35)(30,31)(34,39)(36,40)(37,38);(1,3)(2,4)(5,7)(6,8)(17,19,22,20)(18,23,21,24)(26,31)(29,33)(32,38)(36,39);(2,6)(3,7)(9,10,12,13)(11,15,16,14)(17,20,22,19)(18,23,21,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,11,12,16)(10,14,13,15)(17,21,22,18)(19,24,20,23)(25,28,30,37)(26,32,31,38)(27,34,35,40)(29,36,33,39);(1,4)(2,3)(5,8)(6,7)(25,29)(26,27)(28,36)(30,33)(31,35)(32,34)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fgh,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4861540);