Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,13,16)(11,15,12,14)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,2,5,4)(3,6,7,8)(11,12)(14,15)(19,24)(21,23)(26,30)(31,33)(32,36)(38,39);(9,11,13,12)(10,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,4,5,2)(3,8,7,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=h,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,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, 2703129);