Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,15,16,13)(17,18,21,19)(20,23,24,22)(25,26,29,31)(27,28,30,32);(2,3)(6,7)(9,10,12,14)(11,13,16,15)(17,18,21,19)(20,22,24,23)(25,27)(26,28)(29,30)(31,32);(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22)(25,28)(26,30)(27,31)(29,32);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,24)(18,22,19,23);(1,4,5,8)(2,6,3,7)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=gi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3487618);