Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,12)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26,29,30)(27,31,32,28);(1,3,5,4)(2,6,8,7)(11,15)(14,16)(17,19,21,24)(18,22,23,20)(27,32)(28,31);(1,3,5,4)(2,7,8,6)(9,10,13,12)(11,14,15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28)(29,32)(30,31);(1,4,5,3)(2,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32);(1,2)(3,7)(4,6)(5,8)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ghi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2958008);