Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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)(3,8)(4,7)(5,6)(10,12)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,28)(29,32)(30,31);(1,3,6,4)(2,7,5,8)(11,15)(14,16)(19,24)(21,23)(25,27)(26,30)(28,32)(29,31);(2,5)(7,8)(9,10,13,12)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,30)(26,31,32,27);(1,4,6,3)(2,7,5,8)(9,11,13,15)(10,14,12,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,28)(29,31)(30,32);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,16,15,14)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fghi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3764079);