Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,16,13,15)(18,21)(19,20)(26,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,15,12,16)(17,18,22,21)(19,23,20,24)(25,26,29,28)(27,30,31,32);(3,8)(5,7)(11,13)(15,16)(17,19,22,20)(18,23,21,24)(25,27)(26,30)(28,32)(29,31);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,16,13,15)(17,20,22,19)(18,23,21,24);(1,3,4,8)(2,5,6,7)(9,13,14,11)(10,16,12,15)(17,21,22,18)(19,23,20,24)(25,28,29,26)(27,30,31,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgi,c^2=g,d^2=gh,e^2=f,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, 937641);