Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,4)(6,7)(10,14)(12,16)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(1,2,5,8)(3,6,4,7)(10,14)(11,15)(26,28)(27,34)(32,36)(35,40);(1,3,5,4)(2,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,19,21,18)(20,23,24,22)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,26)(27,33,34,31)(29,32,37,36)(35,38,40,39);(1,4,5,3)(2,7,8,6)(9,12)(10,11)(13,16)(14,15)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=fghi,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2548326);