Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,12,16,15)(17,18,22,23)(19,24,21,20)(27,33)(28,35)(30,36)(31,38);(2,5)(3,8)(9,10,13,14)(11,15,16,12)(18,23)(19,21)(27,33)(30,36)(34,39)(37,40);(3,8)(6,7)(10,14)(11,16)(17,19,22,21)(18,24,23,20)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,3,4,8)(2,6,5,7)(9,11)(10,12)(13,16)(14,15)(17,20,22,24)(18,19,23,21)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,19)(18,24,23,20)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6018124);