Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(1,3,5,6)(2,4,7,8)(9,11,13,12)(10,14,15,16)(26,28)(27,34)(32,36)(35,40);(9,10)(11,16)(12,14)(13,15)(17,19,21,18)(20,23,24,22)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,12,13,11)(10,14,15,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,26)(27,33,34,31)(29,32,37,36)(35,38,40,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,16)(13,15)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2542715);