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 | |
1424 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)(9,10,13,14)(11,15,16,12)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,38)(39,40);(1,2,5,4)(3,6,7,8)(10,14)(11,16)(17,18,21,24)(19,22,20,23)(26,31)(27,35)(33,36)(34,40);(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27,30,35)(26,32,31,37)(28,34,38,40)(29,36,39,33);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,20)(18,22,24,23)(25,28)(26,33)(27,34)(29,32)(30,38)(31,36)(35,40)(37,39);(1,4,5,2)(3,8,7,6)(9,12,13,15)(10,16,14,11)(17,20,21,19)(18,23,24,22)(25,29)(26,34)(27,36)(28,37)(30,39)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=fg,d^2=fh,e^2=fghi,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, 2637703);