Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #759288 |
| 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 | |
2220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,8)(10,11)(12,16)(19,20)(23,24)(25,26)(27,32)(28,30)(29,31);(1,2,5,6)(3,4,7,8)(9,10,13,11)(12,14,16,15)(17,18,22,21)(19,23,20,24)(25,27)(26,28)(29,30)(31,32);(2,6)(4,8)(9,10,13,11)(12,15,16,14)(17,19,22,20)(18,23,21,24)(25,28,29,32)(26,30,31,27);(1,3,5,7)(2,4,6,8)(9,11,13,10)(12,15,16,14)(17,20,22,19)(18,24,21,23)(25,27)(26,28)(29,30)(31,32);(1,4)(2,7)(3,6)(5,8)(9,12,13,16)(10,14,11,15)(17,21,22,18)(19,24,20,23)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 670050);