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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1895 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)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(17,18,22,24)(19,20,21,23)(27,31)(28,32);(1,2,5,4)(3,6,8,7)(10,14)(11,15)(17,19)(18,23)(20,24)(21,22)(25,26,29,30)(27,32,31,28);(2,4)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20,22,23)(18,21,24,19)(25,27)(26,28)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,21,22,19)(18,20,24,23)(25,26)(27,28)(29,30)(31,32);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,14)(13,16)(17,20)(18,21)(19,24)(22,23)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fghi,d^2=fghi,e^2=fgh,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, 3453541);