Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1418 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)(9,10,13,14)(11,12,15,16)(25,26,30,33)(27,34,37,31)(28,38,32,29)(35,36,39,40);(2,4)(6,7)(10,14)(12,16)(18,21)(22,24)(25,27,30,37)(26,31,33,34)(28,39,32,35)(29,40,38,36);(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,15)(14,16)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,14)(13,16)(17,19,20,23)(18,22,21,24)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2861502);