Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(26,27)(28,35)(29,37)(39,40);(2,3)(6,7)(10,13)(14,16)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,39,38,40);(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,39,38,40);(1,3,5,2)(4,6,8,7)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(1,4,5,8)(2,6,3,7)(25,29,30,37)(26,32,27,34)(28,36,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=g,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3388146);