Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(9,10)(11,16)(12,14)(13,15)(18,21)(22,23)(25,26)(27,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,14)(10,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,28)(27,31,32,30);(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,19,20,24)(18,22,21,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|b^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4330777);