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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1985 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,24)(19,21,20,23)(25,26,28,30)(27,29,31,32);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,15,12,16)(17,18)(19,21)(20,23)(22,24)(27,31)(29,32);(3,5)(7,8)(9,10,14,12)(11,16,13,15)(17,19,22,20)(18,23,24,21)(25,26)(27,29)(28,30)(31,32);(1,4,6,2)(3,7,5,8)(9,12,14,10)(11,15,13,16)(17,20,22,19)(18,23,24,21)(25,27)(26,29)(28,31)(30,32);(1,5,6,3)(2,7,4,8)(9,13,14,11)(10,15,12,16)(17,21,22,23)(18,19,24,20)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fgh,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, 2180958);