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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1502 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)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18,21,23)(19,20,24,22)(26,30)(27,33)(28,36)(39,40);(1,3)(2,4)(5,8)(6,7)(17,19)(18,22)(20,23)(21,24)(25,26,29,30)(27,31,33,34)(28,37,36,32)(35,40,38,39);(1,4,6,5)(2,7,8,3)(9,10,13,12)(11,14,15,16)(17,19,21,24)(18,20,23,22)(25,26)(27,34)(28,37)(29,30)(31,33)(32,36)(35,39)(38,40);(1,5,6,4)(2,3,8,7)(9,11)(10,14)(12,16)(13,15)(17,20,21,22)(18,19,23,24)(25,27)(26,31)(28,38)(29,33)(30,34)(32,40)(35,36)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,16,15,14)(17,19)(18,20)(21,24)(22,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,39)(33,38)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fghi,d^2=fg,e^2=hi,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, 4210429);