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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,13)(14,15)(17,18,20,22)(19,21,23,24)(25,26,29,30)(27,31,32,28);(1,2,4,6)(3,5,8,7)(9,10,12,13)(11,15,16,14)(19,23)(21,24)(27,32)(28,31);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,14,13,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,16)(13,14)(17,19,20,23)(18,21,22,24)(25,28)(26,27)(29,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=fgh,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, 2997399);