Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1430 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)(3,4)(5,6)(7,8)(10,14)(11,12)(17,18)(19,22)(20,21)(23,24)(25,26,30,31)(27,28,35,32)(29,33,38,39)(34,36,40,37);(2,6)(3,8)(9,10,13,14)(11,16,12,15)(17,18,21,20)(19,23,24,22)(26,31)(28,32)(33,39)(36,37);(1,3)(2,7)(4,6)(5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,20,21,18)(19,23,24,22)(25,28,30,32)(26,27,31,35)(29,37,38,36)(33,40,39,34);(1,4)(2,8)(3,6)(5,7)(9,12,13,11)(10,15,14,16)(17,19)(18,23)(20,22)(21,24)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,c^2=ghi,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835887);