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 | |
2074 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)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(18,23)(21,24)(26,28)(30,32);(1,2,4,6)(3,7,8,5)(9,11,13,12)(10,14,15,16)(17,18)(19,21)(20,24)(22,23)(27,31)(30,32);(3,8)(5,7)(10,15)(14,16)(17,19,22,20)(18,24,23,21)(25,26,29,28)(27,30,31,32);(1,2)(3,7)(4,6)(5,8)(9,12,13,11)(10,16,15,14)(17,20,22,19)(18,21,23,24)(25,27,29,31)(26,30,28,32);(1,3,4,8)(2,5,6,7)(9,10)(11,16)(12,14)(13,15)(17,21)(18,20)(19,23)(22,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|b^2=fh,c^2=ghi,d^2=gh,e^2=fgh,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, 3846139);