Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,11)(14,15)(19,23)(21,24)(25,26)(27,33)(28,36)(29,34)(30,31)(32,38)(35,39)(37,40);(1,2,5,4)(3,6,7,8)(17,18,20,22)(19,21,23,24)(26,31)(27,34)(32,36)(35,40);(9,10,13,11)(12,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(25,28,30,38)(26,32,31,36)(27,35,34,40)(29,39,33,37);(1,4,5,2)(3,8,7,6)(9,12,13,16)(10,14,11,15)(17,19)(18,21)(20,23)(22,24)(25,29)(26,27)(28,37)(30,33)(31,34)(32,40)(35,36)(38,39)Polycyclic presentation:
2^9|b^2=fghi,c^2=fg,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4356060);