Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,12,13)(11,15,16,14)(17,18,21,22)(19,20,24,23)(25,26,29,30)(27,31,32,28);(2,4)(6,7)(9,10)(11,15)(12,13)(14,16)(18,22)(20,23)(25,27,29,32)(26,28,30,31);(3,8)(6,7)(10,13)(14,15)(17,19,21,24)(18,23,22,20)(25,26,29,30)(27,31,32,28);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=fghi,c^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1034648);