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 | |
1802 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,21,20,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,39)(38,40);(4,8)(6,7)(17,19,22,20)(18,21,23,24)(26,27)(28,35)(29,37)(38,39);(2,3)(6,7)(18,23)(19,20)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,19)(18,21,23,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,21)(18,19)(20,23)(22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fh,c^2=f,d^2=g,e^2=fgi,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, 5067777);