Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 4194304 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2646 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)(18,22)(19,25)(20,27)(31,32);(6,8)(11,16)(13,15)(17,18,21,22)(19,23,25,30)(20,28,27,24)(26,32,29,31);(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,29)(21,25)(22,30)(24,32)(26,27)(28,31);(1,2,3,4)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,30)(26,32)(29,31);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,26)(21,27)(22,28)(23,31)(25,29)(30,32)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=gi,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, 4007555);