Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,18,21,22)(19,23,24,20)(25,26)(27,32)(28,37)(29,30)(31,34)(33,40)(35,39)(36,38);(2,5)(6,8)(18,22)(19,24)(25,27,31,28)(26,32,34,37)(29,38,33,35)(30,36,40,39);(3,7)(6,8)(9,10,13,12)(11,14,15,16)(17,19,21,24)(18,23,22,20)(25,28,31,27)(26,32,34,37)(29,38,33,35)(30,39,40,36);(1,2)(3,8)(4,5)(6,7)(9,11,13,15)(10,14,12,16)(17,18)(19,23)(20,24)(21,22)(25,29,31,33)(26,30,34,40)(27,35,28,38)(32,39,37,36);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,20,21,23)(18,19,22,24)(25,30)(26,33)(27,36)(28,39)(29,34)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fghi,d^2=fgi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3720646);