Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,8)(3,4,6,5)(9,10)(11,14)(12,13)(15,16)(19,23)(21,24)(25,26,30,33)(27,31,28,35)(29,32,37,40)(34,38,36,39);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(25,27,30,28)(26,31,33,35)(29,36,37,34)(32,39,40,38);(1,4,7,5)(2,3,8,6)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,23,24)(25,28,30,27)(26,31,33,35)(29,36,37,34)(32,38,40,39);(1,5,7,4)(2,3,8,6)(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,21,22,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,39)(37,40);(1,6,7,3)(2,4,8,5)(9,10)(11,14)(12,13)(15,16)(25,29,30,37)(26,32,33,40)(27,34,28,36)(31,38,35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fi,d^2=ghi,e^2=gh,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, 5243585);