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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1476 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(19,23)(21,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,5)(3,7)(9,11,14,12)(10,15,13,16)(17,18,20,22)(19,21,23,24)(28,37)(32,36)(34,40)(38,39);(2,5)(6,8)(11,12)(15,16)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,19)(18,21)(20,23)(22,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=fgh,d^2=fg,e^2=gh,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, 4389707);