Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,7,5,4)(9,10,13,12)(11,15,16,14)(18,19)(20,24)(26,28)(27,31);(1,3,6,5)(2,7,8,4)(17,18,21,19)(20,23,24,22)(25,26,30,28)(27,32,31,29);(2,8)(4,7)(9,11,13,16)(10,14,12,15)(17,18,21,19)(20,22,24,23)(25,27)(26,29)(28,32)(30,31);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,18)(20,23,24,22)(25,28,30,26)(27,32,31,29);(1,5,6,3)(2,4,8,7)(9,12,13,10)(11,15,16,14)(17,20,21,24)(18,22,19,23)(25,29,30,32)(26,31,28,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=hi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4883432);