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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1746 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,18,21,20)(19,23,24,22)(26,30)(27,33)(28,35)(38,40);(1,2,4,6)(3,5,7,8)(10,15)(11,12)(26,30)(27,33)(32,37)(34,39);(3,7)(5,8)(9,11,14,12)(10,16,15,13)(17,18,21,20)(19,22,24,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,8)(4,6)(5,7)(9,12,14,11)(10,13,15,16)(17,19,21,24)(18,22,20,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3,4,7)(2,5,6,8)(9,13)(10,12)(11,15)(14,16)(17,20,21,18)(19,23,24,22)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=hi,d^2=ghi,e^2=fghi,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, 3764059);