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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,13,14)(11,15,16,12)(18,20)(21,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(10,14)(12,15)(17,18,22,20)(19,24,23,21)(28,37)(31,36)(34,40)(38,39);(9,11,13,16)(10,15,14,12)(17,19,22,23)(18,21,20,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,18)(19,21,23,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,16,14,11)(17,21)(18,23)(19,20)(22,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fg,d^2=gi,e^2=fgi,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, 3178398);