Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
2310 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)(6,8)(9,10,14,12)(11,15,13,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,31)(29,32);(1,2,4,3)(9,11,14,13)(10,15,12,16)(18,23)(20,30)(24,32)(26,27);(2,3)(9,12,14,10)(11,15,13,16)(17,19)(18,24)(20,27)(21,25)(22,29)(23,32)(26,30)(28,31);(5,6,7,8)(9,13,14,11)(10,15,12,16)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(1,3,4,2)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2216705);