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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1964 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)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,23,24,20)(26,30)(28,32);(1,2,4,6)(3,5,8,7)(11,15)(13,16)(19,24)(20,23)(25,26,29,30)(27,28,31,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,31)(26,28,30,32);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,23)(22,24)(25,28,29,32)(26,31,30,27);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,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, 2997397);