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) = C25×C4 |
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 | |
1420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,5,4,7)(9,10)(11,15)(12,13)(14,16)(19,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(1,3)(2,7)(4,6)(5,8)(9,11,13,16)(10,14,12,15)(26,31)(29,32)(33,39)(37,40);(2,8)(3,4)(10,12)(14,15)(17,18,20,22)(19,21,23,24)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,37,39,40);(1,4,6,3)(2,5,8,7)(9,12,13,10)(11,14,16,15)(17,19)(18,21)(20,23)(22,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,40)(34,36);(1,5)(2,3)(4,8)(6,7)(9,11)(10,15)(12,14)(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=gh,c^2=fi,d^2=g,e^2=fg,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, 4393648);