Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | #15320 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2248 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)(6,8)(9,10,13,11)(12,14,16,15)(17,18,20,21)(19,22,24,23)(25,26)(27,28)(29,32)(30,31);(10,11)(14,15)(17,18)(19,23)(20,21)(22,24)(25,27,29,31)(26,30,32,28);(12,16)(14,15)(18,21)(22,23)(25,28,29,30)(26,31,32,27);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,20,24)(18,22,21,23)(25,27)(26,28)(29,31)(30,32);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=gh,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4317175);