Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2014 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)(3,8)(4,7)(5,6)(10,13)(11,15)(17,18,22,24)(19,20,23,21);(1,2,6,5)(3,8,4,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,20,24,21)(26,29)(30,32);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,19,24,23)(25,26,28,29)(27,30,31,32);(1,4,6,3)(2,7,5,8)(9,11,12,15)(10,14,13,16)(17,18)(19,20)(21,23)(22,24);(1,5,6,2)(3,8,4,7)(9,10)(11,16)(12,13)(14,15)(17,21,22,20)(18,23,24,19)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fgh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1182796);