Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| 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 | #15123 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2507 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)(10,13)(14,15)(17,18,21,23)(19,22,24,20)(25,26)(27,28)(29,31)(30,32);(2,4)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,24)(18,22,23,20)(25,27,29,32)(26,30,31,28);(3,8)(6,7)(10,13)(14,15)(18,23)(19,24)(27,32)(28,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,27)(29,30)(31,32);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,20,21,22)(18,24,23,19)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4862840);