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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1931 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)(25,26,29,30)(27,33,34,31)(28,32,38,37)(35,40,36,39);(4,8)(6,7)(9,11,13,16)(10,12,14,15)(17,18)(19,23)(20,22)(21,24)(26,30)(27,34)(32,37)(35,36);(2,3)(6,7)(10,14)(12,15)(17,18,20,22)(19,21,24,23)(25,27,29,34)(26,31,30,33)(28,36,38,35)(32,40,37,39);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,38)(26,32,30,37)(27,35,34,36)(31,39,33,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,33)(28,37)(29,30)(31,34)(32,38)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=h,d^2=fgh,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, 4078722);