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) = C25×C4 |
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 | |
1628 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)(3,7)(11,15)(14,16)(19,23)(20,22)(25,26,30,31)(27,28,35,32)(29,33,37,39)(34,36,40,38);(3,7)(6,8)(17,18,21,24)(19,20,23,22)(26,31)(28,32)(29,37)(34,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,15,16)(17,19,21,23)(18,22,24,20)(25,27,30,35)(26,32,31,28)(29,34,37,40)(33,38,39,36);(1,3)(2,6)(4,7)(5,8)(9,11,13,15)(10,14,12,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,38,37,36)(33,40,39,34);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20,21,22)(18,23,24,19)(25,29)(26,33)(27,34)(28,36)(30,37)(31,39)(32,38)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fgi,d^2=fghi,e^2=gi,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, 3218294);