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) = C24×C42 |
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 | |
1913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10,13,11)(12,15,16,14)(18,20)(22,23)(25,26)(27,31)(28,32)(29,30);(1,3,5,8)(2,6,7,4)(10,11)(14,15)(18,20)(22,23)(25,27,30,28)(26,31,29,32);(2,7)(3,8)(9,10,13,11)(12,15,16,14)(17,18,21,20)(19,22,24,23)(25,26,30,29)(27,32,28,31);(1,4,5,6)(2,3,7,8)(9,11,13,10)(12,14,16,15)(17,19,21,24)(18,22,20,23)(25,28,30,27)(26,31,29,32);(1,3)(2,4)(5,8)(6,7)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,31,28,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=h,e^2=fg,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, 4035630);