Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | #17068 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(18,21)(19,20)(25,26,28,30)(27,29,31,32);(2,5)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18,22,21)(19,24,20,23)(27,31)(29,32);(3,7)(6,8)(10,13)(11,15)(17,19,22,20)(18,23,21,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,15,13,11)(17,20,22,19)(18,23,21,24)(25,27)(26,29)(28,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,13,14,10)(11,16,15,12)(17,21,22,18)(19,23,20,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2180953);