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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,27,22,24)(20,25,30,21)(28,31,32,29);(6,8)(11,16)(13,15)(17,19,23,22)(18,24,26,27)(20,28,30,32)(21,31,25,29);(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,25)(19,28)(21,26)(22,32)(23,30)(24,29)(27,31);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,21,23,25)(18,20,26,30)(19,29,22,31)(24,32,27,28);(9,11)(10,13)(12,16)(14,15)(17,22,23,19)(18,24,26,27)(20,28,30,32)(21,29,25,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=f,e^2=f,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, 3943986);