Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(17,18,20,22)(19,23,24,21)(25,26,30,32)(27,28,31,29);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(25,27,30,31)(26,28,32,29);(1,3)(2,4)(5,8)(6,7)(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,31)(30,32);(1,4,5,6)(2,3,7,8)(25,28,30,29)(26,27,32,31);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,29,30,28)(26,31,32,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=h,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1319568);