Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
2016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(11,15)(14,16)(18,22)(20,24)(25,26)(27,30)(28,31)(29,34)(32,38)(33,36)(35,40)(37,39);(3,7)(6,8)(18,22)(19,23)(26,27)(31,33)(32,34)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,15,16)(17,18)(19,20)(21,22)(23,24)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,31)(27,33)(29,37)(30,36)(32,40)(34,39)(35,38);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,24)(22,23)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)Polycyclic presentation:
2^9|c^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2666405);