Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,13,15)(11,16,14,12)(25,26)(27,32)(28,36)(29,39)(30,40)(31,33)(34,35)(37,38);(9,10)(11,16)(12,14)(13,15)(18,21)(22,24)(25,27,31,28)(26,32,33,36)(29,37,40,34)(30,35,39,38);(1,2,5,4)(3,6,7,8)(9,11,13,14)(10,12,15,16)(17,18,20,21)(19,22,23,24)(25,28,31,27)(26,32,33,36)(29,37,40,34)(30,38,39,35);(1,3)(2,6)(4,8)(5,7)(9,12,13,16)(10,14,15,11)(25,29)(26,30)(27,34)(28,37)(31,40)(32,38)(33,39)(35,36);(1,4,5,2)(3,8,7,6)(9,10)(11,12)(13,15)(14,16)(17,19,20,23)(18,22,21,24)(25,30)(26,29)(27,35)(28,38)(31,39)(32,37)(33,40)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fhi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2666368);