Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,16,12,14)(17,18)(19,24)(20,22)(21,23)(26,28)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,21,22)(18,20,23,24)(25,26,29,28)(27,31,32,30);(3,7)(5,8)(10,15)(11,12)(17,18)(19,20)(21,23)(22,24)(25,27,29,32)(26,30,28,31);(1,2)(3,8)(4,6)(5,7)(9,12,13,11)(10,16,15,14)(17,18)(19,20)(21,23)(22,24)(25,28,29,26)(27,31,32,30);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,22)(19,23)(21,24)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4105755);