Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2022 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)(4,8)(5,6)(10,14)(12,15)(17,18)(19,20)(21,23)(22,24)(26,28)(27,31);(1,3,5,8)(2,4,6,7)(9,10,13,14)(11,15,16,12)(17,19)(18,22)(20,23)(21,24)(26,28)(30,32);(2,6)(4,7)(10,14)(12,15)(18,23)(19,24)(25,26,29,28)(27,30,31,32);(1,3)(2,7)(4,6)(5,8)(9,11,13,16)(10,12,14,15)(17,20,21,22)(18,24,23,19)(25,27,29,31)(26,30,28,32);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|b^2=gh,c^2=fghi,d^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038410);