Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,7)(11,15)(13,16)(19,23)(22,24)(25,26,29,27)(28,31,32,30);(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(18,21)(22,24)(25,27,29,26)(28,31,32,30);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(17,18,20,21)(19,22,23,24);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,15)(14,16)(25,28)(26,30)(27,31)(29,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=fgh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5588698);