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) = C26 |
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 | |
1647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,15)(14,16)(18,22)(20,24)(25,26)(27,29)(28,31)(30,32);(3,8)(5,7)(9,11,13,16)(10,14,15,12)(18,22)(19,23)(26,27)(30,31);(1,3)(2,5)(4,8)(6,7)(9,12,13,14)(10,11,15,16)(17,18,21,22)(19,20,23,24)(25,27,29,26)(28,30,32,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19,21,23)(18,20,22,24);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,23)(19,22)(21,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,d^2=fhi,e^2=fgh,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, 2340865);