Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(11,15)(14,16)(17,18,22,23)(19,24,26,21)(20,28,29,25)(27,30,32,31);(10,13)(14,16)(18,23)(19,26)(20,29)(30,31);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32);(1,2,3,4)(5,7,8,6)(17,20,22,29)(18,25,23,28)(19,27,26,32)(21,30,24,31);(9,11,12,15)(10,14,13,16)(17,21,22,24)(18,26,23,19)(20,30,29,31)(25,32,28,27)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4296854);