Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,6,4)(9,10,12,15)(11,14,16,13)(17,18)(19,22)(20,24)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(1,3)(2,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,20)(18,22,23,24)(26,31)(29,32)(33,39)(37,40);(3,6)(4,8)(9,11,12,16)(10,14,15,13)(17,20,21,19)(18,22,23,24)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,37,39,40);(1,4,5,8)(2,6,7,3)(9,11)(10,14)(12,16)(13,15)(17,18)(19,24)(20,22)(21,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,40)(34,36);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(25,29,30,32)(26,34,31,27)(28,37,38,40)(33,36,39,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fgi,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431680);