Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(6,7)(9,10)(11,12)(13,15)(14,16)(18,23)(19,20)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,18,22,23)(19,21,20,24)(26,28)(27,34)(32,36)(35,40);(17,19,22,20)(18,21,23,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,3,5,2)(4,6,8,7)(9,12,13,14)(10,11,15,16)(17,20,22,19)(18,21,23,24)(25,28,30,26)(27,33,34,31)(29,32,37,36)(35,38,40,39);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,20)(19,23)(22,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=f,e^2=h,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, 4781032);