Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,14,16)(11,13,12,15)(18,23)(21,24)(25,26)(27,28)(29,32)(30,31);(1,2,4,5)(3,8,7,6)(9,11,14,12)(10,15,16,13)(17,18,22,23)(19,24,20,21)(25,27,29,31)(26,30,32,28);(3,7)(6,8)(9,11,14,12)(10,15,16,13)(17,19,22,20)(18,21,23,24)(25,26,29,32)(27,28,31,30);(1,2)(3,8)(4,5)(6,7)(9,12,14,11)(10,13,16,15)(17,20,22,19)(18,24,23,21)(25,28,29,30)(26,31,32,27);(1,3,4,7)(2,6,5,8)(9,13)(10,12)(11,16)(14,15)(17,21,22,24)(18,20,23,19)(25,26)(27,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=fgi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3919976);