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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,5)(6,7)(9,10,14,13)(11,16,12,15)(18,22)(20,23)(25,26,29,32)(27,31,28,30);(1,3,6,4)(2,5,7,8)(9,11,14,12)(10,15,13,16);(2,7)(3,4)(9,10,14,13)(11,15,12,16)(17,18)(19,23)(20,24)(21,22)(25,27,29,28)(26,30,32,31);(1,4,6,3)(2,5,7,8)(9,12,14,11)(10,15,13,16)(17,19,21,24)(18,23,22,20)(25,27,29,28)(26,30,32,31);(1,5)(2,3)(4,7)(6,8)(9,13,14,10)(11,15,12,16)(17,20,21,23)(18,24,22,19)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=fhi,d^2=fgh,e^2=fg,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, 1085596);