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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(9,10,14,15)(11,13,12,16)(17,18,20,22)(19,21,23,24)(25,26,28,30)(27,31,32,29);(1,3)(2,7)(4,8)(5,6)(10,15)(13,16)(19,23)(21,24)(25,26,28,30)(27,29,32,31);(3,5)(7,8)(9,11,14,12)(10,16,15,13)(17,18)(19,21)(20,22)(23,24)(25,27)(26,29)(28,32)(30,31);(1,4,6,2)(3,7,5,8)(9,12,14,11)(10,13,15,16)(17,19)(18,21)(20,23)(22,24)(25,26)(27,29)(28,30)(31,32);(1,5,6,3)(2,8,4,7)(9,13)(10,12)(11,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=fh,d^2=f,e^2=h,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, 3805335);