Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(18,22)(20,23)(26,28)(29,37)(31,35)(33,39);(11,15)(13,16)(18,22)(19,24)(25,26,30,28)(27,31,34,35)(29,36,37,32)(33,40,39,38);(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,24,20)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,22)(21,24)(25,28,30,26)(27,31,34,35)(29,32,37,36)(33,40,39,38);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,23)(18,19,22,24)(25,29,30,37)(26,32,28,36)(27,33,34,39)(31,38,35,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=h,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3140180);