Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21)(25,26)(27,30)(28,32)(29,31);(1,3,5,8)(2,6,7,4)(10,12)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27,29,32)(26,30,31,28);(2,7)(4,6)(9,11,13,16)(10,14,12,15)(17,18,20,22)(19,21,24,23)(25,27)(26,28)(29,32)(30,31);(1,4,5,6)(2,3,7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24)(25,27)(26,28)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,21,22,23)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=fgi,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3209169);