Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(10,14)(12,15)(18,21)(19,23)(26,27)(28,35)(32,34)(36,37);(2,6)(3,8)(9,10)(11,15)(12,16)(13,14)(19,23)(22,24)(25,26,30,27)(28,33,35,31)(29,32,38,34)(36,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,26)(28,33,35,31)(29,34,38,32)(36,40,37,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19,20,23)(18,22,21,24)(25,28,30,35)(26,31,27,33)(29,37,38,36)(32,40,34,39);(1,4,5,7)(2,8,6,3)(9,12,13,15)(10,16,14,11)(25,29,30,38)(26,32,27,34)(28,36,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=fh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802684);