Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| 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 | |
1804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,11)(12,14,16,15)(17,18,22,23)(19,21,24,20)(26,31)(27,32);(3,4)(6,7)(10,11)(14,15)(18,23)(19,24)(25,26)(27,28)(29,32)(30,31);(1,3,5,4)(2,6,8,7)(9,11,13,10)(12,15,16,14)(17,19,22,24)(18,20,23,21)(25,27,30,32)(26,28,31,29);(1,4,5,3)(2,7,8,6)(17,20,22,21)(18,19,23,24)(25,28,30,29)(26,27,31,32);(1,2)(3,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,21,22,20)(18,24,23,19)(25,29,30,28)(26,32,31,27)Polycyclic presentation:
2^9|a^2=h,c^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863177);