Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1982 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)(4,7,8,6)(9,10,13,12)(11,14,16,15)(18,21)(22,24)(25,26,30,31)(27,34,32,28)(29,33,37,39)(35,38,40,36);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,12,15)(19,23)(22,24)(26,31)(27,32)(29,37)(36,38);(2,3)(6,7)(10,12)(14,15)(17,18,20,21)(19,22,23,24)(25,27,30,32)(26,28,31,34)(29,35,37,40)(33,36,39,38);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,37)(39,40);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=i,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5846484);