Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1458 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)(11,15)(12,16)(17,18,20,22)(19,21,23,24)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,39,40,38);(9,10,14,13)(11,12,15,16)(19,23)(21,24)(26,30)(28,37)(31,33)(34,40);(9,11)(10,12)(13,16)(14,15)(17,19,20,23)(18,21,22,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(9,12,14,16)(10,15,13,11)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,16,15,12)(17,18)(19,21)(20,22)(23,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fh,d^2=fhi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4286860);