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 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,12,14)(11,13,15,16)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(2,6)(3,8)(11,15)(13,16)(17,18,20,22)(19,21,23,24)(26,31)(27,32)(29,37)(35,40);(1,2,5,6)(3,7,8,4)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,21,22,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,4,5,7)(2,3,6,8)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fghi,c^2=gi,d^2=fghi,e^2=g,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, 1806827);