Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,8,7)(9,10,12,15)(11,14,16,13)(19,23)(22,24)(27,29)(28,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(25,26,30,32)(27,31,29,28);(9,11)(10,13)(12,16)(14,15)(17,18,21,20)(19,22,23,24)(25,27,30,29)(26,31,32,28);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28)(26,27)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,28,32,31)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fhi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1295266);