Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| 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 | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(9,10)(11,13)(12,15)(14,16)(18,20)(22,23)(25,26,29,32)(27,30,28,31);(1,3,6,4)(2,7,8,5)(9,10,12,15)(11,14,16,13);(1,3,6,4)(2,5,8,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,20)(19,22,24,23)(25,27,29,28)(26,30,32,31);(1,4,6,3)(2,5,8,7)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,22,20,23)(25,26)(27,30)(28,31)(29,32);(1,5)(2,4)(3,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,20,21,18)(19,23,24,22)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=ghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1413156);