Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1938 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)(9,10,13,16)(11,14,12,15)(17,18,22,24)(19,21,23,20)(26,29)(30,32);(2,3)(6,7)(9,11,13,12)(10,14,16,15)(17,19,22,23)(18,20,24,21)(27,31)(30,32);(4,8)(6,7)(17,20,22,21)(18,23,24,19)(25,26)(27,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,21,22,20)(18,23,24,19)(25,27)(26,30)(28,31)(29,32);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,15,16,14)(17,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=h,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2426898);