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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,12,16,15)(17,18)(19,23)(20,22)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(4,8)(6,7)(9,11,13,16)(10,12,14,15)(17,19)(18,23)(20,21)(22,24)(26,29)(31,33)(32,38)(39,40);(2,3)(6,7)(10,14)(12,15)(18,20)(19,24)(25,27,30,35)(26,31,29,33)(28,36,37,34)(32,40,38,39);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,20,22,18)(19,21,24,23)(25,28)(26,32)(27,34)(29,38)(30,37)(31,39)(33,40)(35,36);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,21)(18,19)(20,24)(22,23)(25,29,30,26)(27,31,35,33)(28,32,37,38)(34,40,36,39)Polycyclic presentation:
2^9|a^2=fgh,b^2=hi,c^2=f,d^2=fg,e^2=f,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, 4070550);