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 | C2×#2714 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2819 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)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31);(6,8)(11,16)(13,15)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31);(9,11,12,16)(10,13,14,15)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=f,e^2=fg,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, 3941351);