Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32);(1,2,4,5)(3,6,7,8)(10,11)(12,16)(17,19,21,24)(18,22,23,20)(25,27,30,29)(26,31,28,32);(9,10,13,11)(12,14,16,15)(17,19)(18,20)(21,24)(22,23)(25,27,30,29)(26,32,28,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,28,30,26)(27,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,22)(23,24)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|b^2=fg,c^2=fhi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1980745);