Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
1961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,22)(23,24)(25,26,30,32)(27,28,29,31);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,32)(29,30);(2,5)(6,8)(10,14)(13,15)(18,21)(19,24)(25,28,30,31)(26,29,32,27);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,29,30,27)(26,28,32,31);(1,2)(3,8)(4,5)(6,7)(9,10)(11,15)(12,14)(13,16)(17,21,22,18)(19,23,24,20)(25,28)(26,29)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=fg,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3967003);