Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,14,15)(11,13,12,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,30)(27,31,33,34)(28,32,37,38)(35,39,36,40);(4,8)(6,7)(9,11,14,12)(10,13,15,16)(17,19,21,24)(18,22,20,23)(26,30)(27,33)(32,38)(35,36);(2,3)(6,7)(10,15)(11,12)(18,20)(22,23)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,37)(38,39);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,20,21,18)(19,23,24,22)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,39)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,37)(26,32,30,38)(27,35,33,36)(31,39,34,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5143681);