Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
2054 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)(3,4)(5,6)(7,8)(10,15)(13,16)(18,20)(19,23)(27,29)(31,32);(1,2,5,6)(3,4,8,7)(9,10,14,15)(11,13,12,16)(18,20)(22,24)(25,26,30,28)(27,32,29,31);(2,6)(4,7)(9,11,14,12)(10,16,15,13)(17,18,21,20)(19,22,23,24)(25,27,30,29)(26,31,28,32);(1,3,5,8)(2,7,6,4)(9,12,14,11)(10,13,15,16)(17,19,21,23)(18,22,20,24)(25,28,30,26)(27,32,29,31);(1,4,5,7)(2,3,6,8)(9,13)(10,12)(11,15)(14,16)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,32,28,31)Polycyclic presentation:
2^9|b^2=fgi,c^2=fg,d^2=h,e^2=g,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, 4035660);