Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(25,26,31,33)(27,34,28,29)(30,32,38,40)(35,39,36,37);(9,11)(10,13)(12,16)(14,15)(18,22)(23,24)(25,27,31,28)(26,29,33,34)(30,35,38,36)(32,37,40,39);(1,2,5,4)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,24)(21,22)(25,28,31,27)(26,29,33,34)(30,35,38,36)(32,39,40,37);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,20)(18,23,22,24)(25,29)(26,27)(28,33)(30,37)(31,34)(32,35)(36,40)(38,39);(1,4,5,2)(3,8,7,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,19)(18,24,22,23)(25,30,31,38)(26,32,33,40)(27,35,28,36)(29,37,34,39)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgi,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2441696);