Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,16,12)(19,23)(21,24)(25,26)(27,29)(28,32)(30,31)(33,38)(34,39)(35,36)(37,40);(1,2)(3,5)(4,7)(6,8)(9,11,13,16)(10,12,14,15)(26,31)(29,38)(32,40)(36,39);(2,3)(6,7)(10,14)(12,15)(17,18,20,22)(19,21,23,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,36,40,39);(1,3,5,2)(4,6,8,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,21)(20,23)(22,24)(25,28,30,37)(26,32,31,40)(27,34,33,35)(29,39,38,36);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,14)(13,16)(25,29)(26,33)(27,31)(28,36)(30,38)(32,34)(35,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fgi,d^2=fgh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 3036629);