Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | |
1654 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)(3,8)(4,7)(5,6)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26,30,33)(27,28,34,31)(29,37,38,32)(35,39,40,36);(1,3,6,4)(2,7,5,8)(10,13)(11,15)(18,20)(19,23)(27,34)(28,31)(35,40)(36,39);(9,11)(10,12)(13,16)(14,15)(17,19,21,23)(18,22,20,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,39)(36,37)(38,40);(1,4,6,3)(2,7,5,8)(9,12,14,16)(10,15,13,11)(25,28,30,31)(26,27,33,34)(29,36,38,39)(32,35,37,40);(1,5,6,2)(3,7,4,8)(9,13,14,10)(11,16,15,12)(17,20,21,18)(19,24,23,22)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=hi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1978603);