Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,13)(14,16)(19,23)(21,24)(25,26)(27,33)(28,35)(29,32)(30,31)(34,39)(36,40)(37,38);(1,3,4,7)(2,5,6,8)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(26,31)(28,33)(29,37)(34,40);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(25,27,30,35)(26,28,31,33)(29,34,37,40)(32,36,38,39);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,21,22,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(25,29,30,37)(26,32,31,38)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4417222);