Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| 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 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,13,12)(11,15,16,14)(17,18,21,23)(19,22,24,20)(25,26)(27,29)(28,32)(30,31)(33,38)(34,36)(35,40)(37,39);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,14,12,15)(17,18)(19,20)(21,23)(22,24)(26,31)(27,33)(32,39)(34,35);(2,7)(3,8)(9,11,13,16)(10,14,12,15)(17,19,21,24)(18,20,23,22)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,4)(2,3)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20,21,22)(18,19,23,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363084);