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, cosocle | C24 | C25 |
| prime radical | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2377 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)(3,7)(4,8)(5,6)(10,14)(11,12)(17,18,22,24)(19,21,23,20)(25,26,29,31)(27,32,30,28);(2,6)(7,8)(9,10,13,14)(11,15,12,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,30)(26,28,31,32);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,21)(18,23,24,19)(25,28,29,32)(26,30,31,27);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,15,14,16)(17,21,22,20)(18,23,24,19)(25,27)(26,28)(29,30)(31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,23)(22,24)(25,26)(27,32)(28,30)(29,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgi,c^2=f,d^2=fh,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, 2751352);