Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
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:
(1,2,6,8)(3,4,7,5)(11,15)(13,16)(25,26,30,29)(27,31,35,34)(28,36,38,32)(33,40,37,39);(1,3,6,7)(2,4,8,5)(9,10,14,12)(11,13,15,16)(18,22)(23,24)(26,29)(31,34)(32,36)(39,40);(1,4,6,5)(2,3,8,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,23,20,24)(25,27,30,35)(26,31,29,34)(28,37,38,33)(32,40,36,39);(1,2)(3,4)(5,7)(6,8)(9,12,14,10)(11,16,15,13)(17,19,21,20)(18,23,22,24)(25,28,30,38)(26,32,29,36)(27,33,35,37)(31,39,34,40);(1,5,6,4)(2,7,8,3)(9,13,14,16)(10,15,12,11)(17,20,21,19)(18,24,22,23)(25,29,30,26)(27,34,35,31)(28,36,38,32)(33,40,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1142909);