Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10,14,15)(11,16,12,13)(19,23)(22,24)(26,29)(27,31);(1,2,5,3)(4,6,8,7)(10,15)(11,12)(17,18,21,20)(19,22,23,24)(27,31)(30,32);(9,11,14,12)(10,13,15,16)(17,18,21,20)(19,22,23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,2)(4,6,8,7)(9,12,14,11)(10,13,15,16)(17,19)(18,22)(20,24)(21,23)(25,27,28,31)(26,30,29,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,15)(14,16)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=fghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835928);