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 | |
1354 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)(10,14)(12,16)(25,26,29,30)(27,31,35,33)(28,36,38,32)(34,40,37,39);(10,14)(11,15)(17,18,20,22)(19,21,23,24)(26,30)(31,33)(32,36)(39,40);(1,2,5,4)(3,6,7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,11,13,15)(10,12,14,16)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,4,5,2)(3,8,7,6)(9,12)(10,15)(11,14)(13,16)(17,19,20,23)(18,21,22,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fgi,d^2=fg,e^2=ghi,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, 2665724);