Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
3337 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)(19,24,25,20)(21,29,30,26)(27,28,32,31);(10,13)(14,16)(18,23)(19,25)(21,30)(28,31);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32);(1,2,3,4)(5,7,8,6)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)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,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4250219);