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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(10,14)(12,15)(17,18,22,21)(19,20,24,23)(26,28)(27,31);(1,3)(2,6)(4,5)(7,8)(9,10,13,14)(11,12,16,15)(17,19,22,24)(18,23,21,20)(27,31)(30,32);(3,4)(6,7)(10,14)(12,15)(17,20)(18,19)(21,24)(22,23)(25,26,29,28)(27,30,31,32);(1,2)(3,7)(4,6)(5,8)(9,11,13,16)(10,15,14,12)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,32)(29,31);(1,4,5,3)(2,7,8,6)(9,12,13,15)(10,11,14,16)(17,21,22,18)(19,23,24,20)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,c^2=fgh,d^2=ghi,e^2=gh,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, 3902384);