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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,13,15)(11,12,16,14)(17,18,22,21)(19,24,20,23)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(29,37)(31,38)(34,40)(36,39);(2,7)(3,8)(10,15)(11,16)(17,19,22,20)(18,23,21,24)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,4,5,6)(2,3,7,8)(9,12)(10,11)(13,14)(15,16)(17,20,22,19)(18,23,21,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,21,22,18)(19,23,20,24)(25,29,30,37)(26,31,32,38)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=fi,e^2=gh,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, 5026771);