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 | |
1744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,12)(13,14)(15,16)(18,22)(19,20)(25,26,29,27)(28,31,32,30);(2,5)(6,8)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,24)(21,22)(25,26,29,27)(28,30,32,31);(3,7)(6,8)(10,14)(12,16)(17,19,21,20)(18,23,22,24)(25,27,29,26)(28,31,32,30);(1,2)(3,8)(4,5)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20,21,19)(18,24,22,23)(25,28,29,32)(26,30,27,31);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,24)(20,23)(21,22)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=fgi,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, 3487623);