Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,21,23,24)(28,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(19,23)(21,24)(25,26,29,27)(28,30,32,31);(3,8)(5,7)(10,14)(12,15)(17,19,20,23)(18,21,22,24)(25,27,29,26)(28,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,32)(26,30,27,31);(1,3,4,8)(2,5,6,7)(9,12,13,15)(10,11,14,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=fgi,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4626843);