Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14885 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2689 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,23,24,22)(25,26,30,29)(27,32,28,31);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,24,23)(27,28)(31,32);(2,6)(4,8)(11,16)(13,15)(25,27,30,28)(26,31,29,32);(1,3,5,7)(2,4,6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,28,30,27)(26,32,29,31);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,32,28,31)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=h,d^2=fi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 989900);