Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(1,2,5,3)(4,7,8,6)(10,13)(14,15)(17,19,21,23)(18,22,24,20)(26,30)(31,38)(32,35)(39,40);(2,3)(4,8)(9,11,12,16)(10,14,13,15)(17,20,21,22)(18,23,24,19)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,22)(23,24)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,24)(21,23)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fhi,c^2=fh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4106160);