Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1356 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)(3,4,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,35)(29,32)(30,31)(34,40)(36,38)(37,39);(1,3)(2,4)(5,8)(6,7)(11,15)(13,16)(17,19,21,18)(20,23,24,22)(26,31)(28,33)(29,37)(34,38);(2,6)(3,8)(9,10,12,14)(11,13,15,16)(25,27)(26,28)(29,38)(30,35)(31,33)(32,40)(34,37)(36,39);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,24)(18,22,19,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4,5,7)(2,3,6,8)(9,11)(10,13)(12,15)(14,16)(25,29,30,37)(26,32,31,39)(27,34,35,38)(28,36,33,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=fgh,d^2=fhi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5030218);