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 | |
2518 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)(3,7)(4,5)(6,8)(10,12)(11,15)(17,18,22,20)(19,24,23,21)(25,26)(27,31)(28,29)(30,32);(1,3,5,8)(2,6,4,7)(9,10,13,12)(11,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,30)(28,31)(29,32);(2,4)(6,7)(10,12)(14,16)(19,23)(21,24)(25,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,15)(10,14,12,16)(17,20,22,18)(19,24,23,21);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,16,15,14)(17,21,22,24)(18,23,20,19)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=h,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, 937423);