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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,7)(11,12)(14,15)(25,26)(27,32)(28,29)(30,31)(33,37)(34,40)(35,39)(36,38);(3,7)(6,8)(9,10,13,16)(11,14,12,15)(17,18,21,20)(19,23,24,22)(26,31)(28,36)(32,39)(33,40);(1,2,5,4)(3,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,21,24)(18,22,20,23)(25,27)(26,32)(28,33)(29,37)(30,35)(31,39)(34,38)(36,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,21,20)(19,22,24,23)(25,28,30,36)(26,29,31,38)(27,33,35,40)(32,37,39,34);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,23,24,22)(25,29)(26,28)(27,34)(30,38)(31,36)(32,40)(33,39)(35,37)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,d^2=fh,e^2=gh,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, 3304113);