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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2674 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)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,19)(20,22,24,23)(25,26)(27,31)(28,30)(29,32);(1,2,4,5)(3,6,7,8)(9,10,13,12)(11,14,16,15)(18,19)(22,23)(25,27,30,29)(26,31,28,32);(3,7)(6,8)(10,12)(14,15)(17,18,21,19)(20,23,24,22)(25,26,30,28)(27,32,29,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,23,24,22)(25,28,30,26)(27,31,29,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,21,24)(18,22,19,23)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=fi,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, 2180957);