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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,23)(22,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,3)(2,4)(5,7)(6,8)(17,19,22,21)(18,20,24,23)(26,31)(27,34)(29,37)(36,39);(2,6)(3,7)(9,11,14,13)(10,15,12,16)(17,19,22,21)(18,23,24,20)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,2)(3,4)(5,6)(7,8)(9,12,14,10)(11,15,13,16)(17,20,22,23)(18,19,24,21)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,11)(10,15,12,16)(17,21,22,19)(18,23,24,20)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=gh,d^2=gh,e^2=fgh,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, 4869313);