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 | |
1763 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)(11,12)(13,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26)(27,35)(28,38)(29,37)(30,31)(32,36)(33,39)(34,40);(1,2,4,6)(3,5,7,8)(9,11,14,15)(10,13,16,12)(26,31)(27,36)(33,38)(34,37);(3,7)(5,8)(9,12,14,13)(10,11,16,15)(17,19,21,18)(20,23,24,22)(25,27)(26,32)(28,37)(29,33)(30,36)(31,35)(34,39)(38,40);(1,2)(3,8)(4,6)(5,7)(9,13,14,12)(10,15,16,11)(17,20,21,24)(18,22,19,23)(25,28,30,39)(26,33,31,38)(27,37,36,34)(29,35,40,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,16)(25,29)(26,34)(27,38)(28,32)(30,40)(31,37)(33,36)(35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=hi,d^2=ghi,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, 3460171);