Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,30)(29,32)(33,35)(34,39)(36,38)(37,40);(4,8)(6,7)(9,11,13,16)(10,14,15,12)(17,19,21,24)(18,22,20,23)(26,28)(31,33)(32,36)(39,40);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,20,21,18)(19,22,24,23)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,14)(15,16)(25,28,30,26)(27,33,35,31)(29,32,38,36)(34,39,37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,22)(21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3978437);