Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2810 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)(3,5)(4,7)(6,8)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,22)(21,24)(26,28)(30,31);(2,3)(4,8)(10,12)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,29)(30,32);(4,8)(6,7)(10,12)(14,16)(19,24)(21,23)(26,28)(30,31);(1,3,5,2)(4,6,8,7)(9,11,13,15)(10,14,12,16)(17,18)(19,23)(20,22)(21,24)(25,27,29,32)(26,30,28,31);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,15,16)(17,19,20,24)(18,21,22,23)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,d^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3875003);