Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,15,16,14)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(2,4)(6,7)(18,22)(23,24)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(3,8)(6,7)(9,10,13,12)(11,14,16,15)(17,18,21,22)(19,23,20,24)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,19,21,20)(18,23,22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,24,22,23)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fghi,d^2=fghi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1689333);