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 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,15)(17,18,22,20)(19,24,21,23)(25,26,30,31)(27,29,35,32)(28,33,37,38)(34,36,40,39);(9,10,13,14)(11,12,15,16)(26,31)(29,32)(33,38)(36,39);(2,5)(6,8)(10,14)(11,15)(17,19,22,21)(18,23,20,24)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,18)(19,23,21,24)(25,28,30,37)(26,33,31,38)(27,34,35,40)(29,36,32,39);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,16)(14,15)(17,21,22,19)(18,23,20,24)(25,29,30,32)(26,27,31,35)(28,36,37,39)(33,34,38,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgi,d^2=ghi,e^2=g,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, 4909024);