Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1664 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)(10,13)(11,15)(17,18)(19,20)(21,24)(22,23)(27,28)(30,31);(2,3)(6,7)(10,13)(14,16)(17,19)(18,21)(20,22)(23,24)(25,26,29,32)(27,30,28,31);(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,24,22,19)(25,27,29,28)(26,30,32,31);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,20)(18,24,22,19)(25,26)(27,30)(28,31)(29,32);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,15)(13,16)(17,22,23,18)(19,20,24,21)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fhi,b^2=gh,c^2=gi,d^2=f,e^2=fgh,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, 2149378);